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THE ATOM AND THE BOHR THEORY
OF ITS STRUCTURE

Original Title: “ Bohr s Atomteori , almenfatteligt fremstillet

Translated from the Danish by R. B. Lindsay x
Fellow of the American- Scandinavian Foundation ,

1923, and Rachel T. Lindsay

THE ATOM

AND

THE BOHR THEORY
OF ITS STRUCTURE

An Elementary Presentation

H. A. KRAMERS

I.ECTURER AT THE INSTITUTE OF THEORETICAL PHYSICS IN THE
UNIVERSITY OF COPENHAGEN

AND

HELGE HOLST

LIBRARIAN AT THE ROYAL TECHNICAL COLLEGE OF COPENHAGEN

WITH A FOREWORD BY

SIR ERNEST RUTHERFORD, F.R.S*

NEW YORK

ALFRED A. KNOPF

. I . 'j

PRINTED IN GREAT BRITAIN BY MORRISON AND GIBB LTD., EDINBURGH

PREFACE

At the close of the nineteenth century and the

beginning of the twentieth, our knowledge of the

activities in the interior of matter experienced a

development which surpassed the boldest hopes that
could have been entertained by the chemists and

physicists of the nineteenth century. The smallest
particles of chemistry, the atoms of the elements, which
hitherto had been approached merely by inductive
thought, now became tangible realities, so to speak, which
could be counted and whose tracks could be photo¬
graphed. A series of remarkable experimental investiga¬
tions, stimulated largely by the English physicist, J. J.
Thomson, had disclosed the existence of negatively
charged particles, the so-called electrons, 20Yo the mass
of the smallest atom of the known elements. A theory
of electrons, based on Maxwell’s classical electro¬
dynamical theory and developed mainly through the
labours of Lorentz in Holland and Larmor in England,
had brought the problem of atomic structure into close
connection with the theory of radiation. The experi¬
ments of Rutherford proved, beyond a doubt, that
atoms were composed simply of light, negative electric
particles, and small heavy, positive electric particles.

vii

136802

Vlll

PREFACE

The new " quantum theory ” of Planck was proving
itself very powerful in overcoming grave difficulties in
the theory of radiation. The time thus seemed ripe
for a comprehensive investigation of the fundamental
problem of physics— the constitution of matter, and an
explanation in terms of simple general laws of the
physical and chemical properties of the atoms of the
elements.

During the first ten years of the new century the
problem was attacked with great zeal by many scientists,
and many interesting atomic models were developed and
studied. But most of these had more significance for
chemistry than for physics, and it was not until 1913
that the work of the Danish physicist, Niels Bohr, paved
the way for a really physical investigation of the problem
in a remarkable series of papers on the spectrum and
atomic structure of hydrogen. The ideas of Bohr,
founded as they were on the quantum theory, were
startling and revolutionary, but their immense success
in explaining the facts of experience after a time won
for them the wide recognition of the scientific world, and
stimulated work by other investigators along similar
lines. The past decade has witnessed an enormous
development at the hands of scientists in all parts of the
world of Bohr’s original conceptions ; but through it all
Bohr has remained the leading spirit, and the theory
which, at the present time, gives the most comprehensive
view of atomic structure may, therefore, most properly
bear the name of Bohr.

PREFACE

It is the object of this book to give the reader a
glimpse of the fundamental conceptions of this theory,
together with some of the most significant results it
has attained. The book is designed to meet the needs
of those who wish to keep abreast of modern develop¬
ments in science, but have neither time nor inclination
to delve into the highly mathematical abstract literature
in which the developments are usually concealed. It
is with this in mind that the first four chapters have been
devoted to a general survey of those parts of physics
and chemistry which have close connection with atomic
theory. No attempt has been made at a mathematical
development, and the physical meaning of such mathe¬
matical formulae as do occur has been clearly emphasized
in the text. It is hoped, however, that even those
readers whose acquaintance with atomic theory is more
than casual, will find the book a stimulus to further
study of the Bohr theory.

Here we wish to record our best thanks to Mr. and
Mrs. Lindsay for the ability and the great care with
which they have carried out the translation from the
Danish original.

FOREWORD

During the last decade there has been a great advance
in our knowledge of the structure of the atom and of the
relation between the atoms of the chemical elements.
In the later stages, science owes much to the remarkable
achievements of Professor Niels Bohr and his co-workers
in Copenhagen. For the first time, we have been given
a consistent theory to explain the arrangement and
motion of the electrons in the outer atom. The theory
of Bohr is not only able to account in considerable
detail for the variation in the properties of the elements
exemplified by the periodic law, but also for the main
features of the spectra, both X-ray and optical, shown
by all elements.

This volume, written by Dr. Kramers and Mr. Holst,
gives a simple and interesting account of our knowledge
of atomic structure, with special reference to the work
of Professor Bohr. Dr. Kramers is in an especially
fortunate position to give a first-hand account of this
subject, for he has been a valued assistant to Professor
Bohr in developing his theories, and has himself made
important original contributions to our knowledge in
this branch of inquiry.

xii FOREWORD

I can confidently recommend this book to English
readers as a clearly written and accurate account of the
development of our ideas on atomic structure. It is
written in simple language, and the essential ideas
are explained without mathematical calculations. This
book should prove attractive not only to the general
scientific reader, but also to the student who wishes to
gain a broad general idea of this subject before entering
into the details of the mathematical theory.

E. RUTHERFORD.

Cavendish Laboratory,
Cambridge, 8 th October 1923.

CONTENTS

PAGE

Preface . . . . . . vii

Foreword . . . . . . xi

CHAP.

I. Atoms and Molecules ..... i

II. Light Waves and the Spectrum . . -34

III. Ions and Electrons . . . . .61

IV. The Nuclear Atom . . . . -83

V. The Bohr Theory of the Hydrogen Spectrum . 105

VI. Various Applications of the Bohr Theory .153

VII. Atomic Structure and the Chemical Properties

of the Elements . . . . .180

Interpretation of Symbols and Physical Con¬
stants ....... 209

COLOURED PLATES

PLATE

I. Spectrum Plates according to the Original

Drawings of Bunsen and Kirchhoff . At end

II. Principal Features of Atomic Structure in
Some of the Elements — Atomic Structure
of Radium ..... At end

THE

ATOM AND THE BOHR THEORY
OF ITS STRUCTURE

CHAPTER I
ATOMS AND MOLECULES

Introduction.

As early as 400 B.c. the Greek philosopher, Demo¬
critus, taught that the world consisted of empty space
and an infinite number of small invisible particles. These
particles, differing in form and magnitude, by their
arrangements and movements, by their unions and
disunions, caused the existence of physical bodies with
different characteristics, and also produced the observed
variations in these bodies. This theory, which no doubt
antedated Democritus, later became known as the
Atomic Theory, since the particles were called atoms,
i.e. the “ indivisible.”

But the atomic conception was not the generally
accepted one in antiquity. Aristotle (384-322 b.c.)
was not an atomist, and denied the existence of dis¬
continuous matter ; his philosophy had a tremendous
influence upon the ideas of the ancients, and even upon
the beliefs of the Middle Ages. It must be confessed
that his conception of the continuity of matter seemed
to agree best with experiment, because of the apparent

THE ATOM AND THE BOHR THEORY

homogeneity of physical substances such as metal, glass,
water and air. But even this apparent homogeneity
could not be considered entirely inconsistent with the
atomic theory, for, according to the latter, the atoms
were so small as to be invisible. Moreover, the atomic
theory left the way open for a more complete under¬
standing of the properties of matter. Thus when air was
compressed and thereafter allowed to expand, or when
salt was dissolved in water producing an apparently new
homogeneous liquid, salt water, or when silver was
melted by heat, or light changed colour on passing
through wine, it was clear that something had happened
in the interior of the substances in question. But
complete homogeneity is synonymous with inactivity.
How is it possible to obtain a definite idea of the
inner activity lying at the bottom of these changes
of state, if we do not think of the phenomenon as an
interplay between the different parts of the apparently
homogeneous matter ? Thus, in the examples above,
the decrease in the volume of the air might be considered
as due to the particles drawing nearer to each other ;
the dissolving of salt in water might be looked upon
as the movement of the salt particles in between the
water particles and the combination of the two kinds ;
the melting of silver might naturally appear to be due
to the loosening of bonds between the individual silver
particles.

The atomic theory had thus a sound physical basis,
and proved particularly attractive to those philosophers
who tried to explain the mysterious activity of matter
in terms of exact measurements. The atomic hypo¬
thesis was never completely overthrown, being supported
after the time of Aristotle by Epicurus ( c . 300 b.c.), who

ATOMS AND MOLECULES

introduced the term “atom,” and by the Latin poet,
Lucretius (c. 75 b.c.) in his De Rerum Natura. Even
in the Middle Ages it was supported by men of inde¬
pendent thought, such as Nicholas of Autrucia, who
assumed that all natural activities were due to unions
or disunions of atoms. It is interesting to note that in
1348 he was forced to retract this heresy. With the
impetus given to the new physics by Galileo (1600) the
atomic view gradually spread, sometimes explicitly
stated as atomic theory, sometimes as a background for
the ideas of individual philosophers. Various investi¬
gators developed comprehensive atomic theories in which
they attempted to explain nearly everything from
purely arbitrary hypotheses ; they occasionally arrived
at very curious and amusing conceptions. For example,
about 1650 the Frenchman, Pierre Gassendi, following
some of the ancient atomists, explained the solidity of
bodies by assuming a hook-like form of atom so that the
various atoms in a solid body could be hooked together.
He thought of frost as an element with tetrahedral
atoms, that is, atoms with four plane faces and with four
vertices each ; the vertices produced the characteristic
pricking sensation in the skin. A much more thorough
treatment of the atomic theory was given by Boscowich
(1770). He saw that it was unnecessary to conceive
of the atoms as spheres, cubes, or other sharply defined
physical bodies ; he considered them simply as points in
space, mathematical points with the additional property
of being centres of force. He assumed that any two
atoms influenced each other with a force which varied,
according to a complicated formula, with the distance
between the centres. But the time was hardly ripe for
such a theory, inspired as it evidently was by Newton’s

THE ATOM AND THE BOHR THEORY

teachings about the gravitational forces between the
bodies of the universe. Indeed there were no physical
experiments whose results could, with certainty, be
assumed to express the properties of the individual atoms.

The Atomic Theory and Chemistry.

In the meantime atomic investigations of a very
different nature had been influencing the new science of
chemistry, in which the atomic theory was later to prove

dryness

Fig. i. — The four elements and the four fundamental characteristics.

itself extraordinarily fruitful. It was particularly un¬
fortunate that in chemistry, concerned as it is with the
inner activities of the elements, Aristotle’s philosophy
was long the prevailing one. He adopted and developed
the famous theory of the four “ elements,” namely, the
dry and cold earth, the cold and damp water, the damp
and warm air, the warm and dry fire. These " elements ”
must not be confused with the chemical elements known
at the present day ; they were merely representatives
of the different consistent combinations of the four

ATOMS AND MOLECULES

fundamental qualities, dryness and wetness, heat and
cold. From the symmetry in the system these were
supposed to be the principles by means of which all the
properties of matter could be explained. Neither the
four " elements ” nor the four fundamental qualities
could be clearly defined ; they were vague ideas to
be discussed in long dialectic treatises, but were
founded upon no physical quantities which could be
measured.

A system of chemistry which had its theoretical
foundations in the Greek elemental conceptions naturally
had to work in the dark. Undoubtedly this uncertainty
contributed to the relatively insignificant results of all
the labour expended in the Middle Ages on chemical
experiments, many of which had to do with the attempt
to transmute the base metals into gold. Naturally there
were many important contributions to chemistry, and
the theories were changed and developed in many ways
in the course of time. The alchemists of the Middle
Ages thought that metal consisted only of sulphur and
quicksilver ; but the interpretation of this idea was
influenced by the Greek elemental theory which was
maintained at the same time ; thus these new metal
“ elements ” were considered by many merely as the
expressions of certain aspects of the metallic charac¬
teristics, rather than as definite substances, identical
with the elements bearing these names. It is, however,
necessary to guard against attributing to a single con¬
ception too great influence on the historical development
of the chemical and physical sciences. That the growth
of the latter was hindered for so long a time was due more
to the uncritical faith in authority and to the whole
characteristic psychological point of view which governed

THE ATOM AND THE BOHR THEORY

Western thought in the centuries preceding the Renais¬
sance.

Robert Boyle (1627-1691) is one of the men to whom
great honour is due for brushing aside the old ideas
about the elements which had originated in obscure
philosophical meditations. To him an element was
simply a substance which by no method could be
separated into other substances, but which could unite
with other elements to form chemical compounds
possessing very different characteristics, including that
of being decomposable into their constituent elements.
Undoubtedly Boyle’s clear conception of this matter
was connected with his representation of matter as of an
atomic nature. According to the atomic conception,
the chemical processes do not depend upon changes
within the element itself, but rather in the union or
disunion of the constituent atoms. Thus when iron
sulphide is produced by heating iron and sulphur
together, according to this conception, the iron atoms
and the sulphur atoms combine in such a way that each
iron atom links itself with a sulphur atom. There is
then a definite meaning in the statement that iron
sulphide consists of iron and sulphur, and that these two
substances are both represented in the new substance.
There is also a definite meaning, for instance, in the
statement that iron is an element, namely, that by no
known means can the iron be broken down into different
kinds of atoms which can be reunited to produce a
substance different from iron.

The clarity which the atomic interpretation gave to
the conception of chemical elements and compounds
was surely most useful to chemical research in the follow¬
ing years ; but before the atomic theory could play a

ATOMS AND MOLECULES

really great role in chemistry, it had to undergo consider¬
able development. In the time of Boyle, and even later,
there was still uncertainty as to which substances were
the elements. Thus, water was generally considered
as an element. According to the so-called phlogiston
theory developed by the German Stahl (1660-1734),
a theory which prevailed in chemistry for many years,
the metals were chemical compounds consisting of a
gaseous substance, phlogiston, which was driven off
when the metals were heated in air, and the metallic
oxide which was left behind. It was not until the latter
half of the eighteenth century that the foundation was
laid for the new chemical science by a series of dis¬
coveries and researches carried on by the Swedish
scientist Scheele, the Englishmen Priestley and Caven¬
dish, and particularly by the Frenchman Lavoisier. It
was then discovered that water is a chemical compound
of the gaseous elements oxygen and hydrogen, that air
is principally a mixture (not a compound) of oxygen and
nitrogen, that combustion is a chemical process in which
some substance is united with oxygen, that metals are
elements, while metallic oxides, on the other hand, are
compounds of metal and oxygen, etc. Of special
significance for the atomic theory was the fact that
Lavoisier made weighing one of the most powerful tools
of scientific chemistry.

Weighing had indeed been used previously in chemical
experiments, but the experimenters had been satisfied
with very crude precision, and the results had little
influence on chemical theory. For example, the phlo¬
giston theory was maintained in spite of the fact that it
was well known that metallic oxide weighed more than
the metal from which it was obtained. Lavoisier now

THE ATOM AND THE BOHR THEORY

showed, by very careful weighings, that chemical com¬
binations or decompositions can never change the total
weight of the substances involved ; a given quantity of
metallic oxide weighs just as much as the metal and the
oxygen taken individually, or vice versa. From the
point of view of the atomic theory, this obviously means
that the weight of individual atoms is not changed in
the combinations of atoms which occur in the chemical
processes. In other words, the weight of an atom is an
invariable quantity. Here, then, we have the first pro¬
perty of the atom itself to be established by experiment
— a property, indeed, which most atomists had already
tacitly assumed.

Moreover, by the practice of weighing it was deter¬
mined that to every chemical combination there corresponds
a definite weight ratio among the constituent parts. This
also had been previously accepted by most chemists as
highly probable ; but it must be admitted that the law
at one time was assailed from several sides.

In comparing the weight ratios in different chemical
compounds certain rules were, in the meantime, obtained.
In many ways the most important of these, the so-called
law of multiple proportions, was enunciated in the begin¬
ning of the last century by the Englishman, John
Dalton. As an example of this law we may take two
compounds of carbon and hydrogen called methane or
marsh gas and ethylene, in which the quantities of
hydrogen compounded with the same quantity of carbon
are as two is to one. Another example may be seen
in the compounds of carbon and oxygen. In the two
compounds of carbon and oxygen, carbon monoxide and
carbon dioxide, the weight ratios between the carbon
and oxygen are respectively as three to four and three

ATOMS AND MOLECULES

to eight. A definite quantity of carbon has thus in carbon
dioxide combined with just twice as much oxygen as in
carbon monoxide. No less than five oxygen com¬
pounds with nitrogen are known, where with a given
quantity of nitrogen the oxygen is combined in ratios of
one, two, three, four and five.

These simple number relations can be explained very
easily by the atomic theory, by assuming, first, that all
atoms of the same element have the same weight ; and
second, that in a chemical combination between two
elements the atoms combine to form an atomic group
characteristic of the compound in question — a compound
atom , as Dalton called it, or a molecule , as the atomic
group is now called. These molecules consist of com¬
paratively few atoms, as, for example, one of each kind,
or one of one kind and two, three or four of another, or
two of one kind and three or four of another, etc. When
three elements are involved in a chemical compound the
molecule must contain at least three atoms, but there
may be four, five, six or more. The law of multiple
proportions thus takes on a more complicated character,
but it remains apparent even in this case.

When Dalton in the beginning of last century formu¬
lated the theory of the formation of chemical compounds
from the atoms of the elements, he at once turned atomic
theory into the path of more practical research, and it
was soon evident that chemical research had then ob¬
tained a valuable tool. It may be said that Dalton’s
atomic theory is the firm foundation upon which modern
chemistry is built.

While Dalton’s theory could not give information
about the absolute weights in grams of the atoms of
various elements, it could say something about the

10 THE ATOM AND THE BOHR THEORY

relative atomic weights, i.e., the ratios of the weights of
the different kinds of atoms, although it is true that these
ratios could not always be determined with certainty.
If, for example, the ratio between the oxygen and hydro-

T 9

tTiryUe

9 10

0®

JBi/ntury

23 v

(DO

35 16

Ternary

26 27 28

(DOG) CKDO

Fig. 2. — For description, see opposite page.

ATOMS AND MOLECULES

gen in water is found to be as eight to one, then the
weight ratio between the oxygen atom and the hydrogen
atom will be as eight to one, if the water molecule is
composed of one oxygen atom and one hydrogen atom
(as Dalton supposed, see Fig. 2). But it will be as sixteen
to one, if the water molecule is composed of one oxygen
and two hydrogen atoms (as we now know to be the case).
On the other hand, a ratio of seven to one will be com¬
patible with the experimental ratio of eight to one only
if we assume that the water molecule consists of fifteen
atoms, eight of oxygen and seven of hydrogen, a very
improbable hypothesis. In another case let us compare
the quantities of oxygen and of hydrogen which are
compounded with the same quantities of carbon in the
two substances, carbon monoxide and methane respec¬
tively. On the assumption that the molecules in question
have a simple structure, we can draw conclusions about
the ratio of the atomic weights of hydrogen and oxygen.
Now, if a ratio such as seven to one or fourteen to one is

Fig. 2. — Representation of a part of Dalton’s atomic table (of 1808)
where the atom of each element has its own symbol, and
chemical compounds are indicated by the union of the atoms
of the elements into groups by 2, 3, 4 . . . (binary, ternary,
quaternary . . . atoms). Below are given the designations of
the different atoms, and in parentheses the atomic weight given
by Dalton with that of hydrogen as unity and the designations
of the indicated atomic groups.

Atoms of the Elements.- — 1. Hydrogen (1) ; 2. Azote (5) ; 3. Carbon
(5) ; 4. Oxygen (7) ; 5. Phosphorus (9) ; 6. Sulphur (13) ; 7. Mag¬
nesia (20) ; 8. Lime (23) ; 9. Soda (28) ; 10. Potash (42) ; 11. Strontites
(46) ; 12. Barytes (68) ; 13. Iron (38) ; 14. Zinc (56) ; 15. Copper (56) ;
16. Lead (95) ; 17. Silver (100) ; 18. Platina (100) ; 19. Gold (140) ;
20. Mercury (167).

Chemical Compounds. — 21. Water; 22. Ammonia; 23. 26. 27. and
30. Oxygen compounds of Azote ; 24. 29. and 33. Hydrogen com¬
pounds of Carbon; 25. Carbon monoxide; 28. Carbon dioxide; 31.
Sulphuric acid ; 32. Hydro-sulphuric acid.

12 THE ATOM AND THE BOHR THEORY

obtained while the analysis of water gives eight to one
or sixteen to one, then either the structure of the mole¬
cule is more complicated than was assumed, or the
analyses must be improved by more careful experiments.
We can thus understand that the atomic theory can
serve as a controlling influence on the analysis of chemical
compounds.

In order to choose between the different possible
ratios of atomic weights, for example, the eight to one
or the sixteen to one in the case of oxygen and hydrogen,
Dalton had to make certain arbitrary assumptions.
The first of these is that two elements of which only one
compound is known appear with but one atom each in
a molecule. Partly on account of this assumption and
partly on account of the incompleteness of his analyses,
Dalton’s values of the ratios of the atomic weights of
the atoms and his pictures of the structure of molecules
differ from those of the present day, as is obvious from
Fig. 2.

A much firmer foundation for the choice made
appears later in the Avogadro Law, starting with the
fact that different gases show great similarity in their
physical conduct — for instance, all expand by an in¬
crease of 1/273 of their volume with an increase in tem¬
perature from o° C. to i° C. — the Italian, Avogadro, in
1811, put forward the hypothesis that equal volumes of
all gases at the same temperature and pressure contain
equal numbers of molecules. A few examples suffice
to show the usefulness of this rule.

When one volume of the gas chlorine unites with one
volume of hydrogen there result two volumes of the gas,
hydrogen chloride, at the same temperature and pres¬
sure. According to Avogadro’s Law one molecule of

ATOMS AND MOLECULES

chlorine and one molecule of hydrogen unite to become
two molecules of hydrogen chloride, and since each of
these two molecules must contain at least one atom of
hydrogen and one atom of chlorine, it follows that
one molecule of chlorine must contain two atoms of
chlorine and that one molecule of hydrogen must con¬
tain two atoms of hydrogen. From this one can see
that even in the elements the atoms are united into
molecules. It is now well established that most ele¬
ments have diatomic molecules, though some, including
mercury and many other metals, are monatomic. When
oxygen and hydrogen unite to form water, one litre of
oxygen and two litres of hydrogen produce two litres of
water vapour at same temperature and pressure. Ac¬
cordingly, one molecule of oxygen and two molecules
of hydrogen form two molecules of water. If the
oxygen molecule is diatomic like the hydrogen, then one
molecule of water contains one atom of oyxgen and two
atoms of hydrogen. Since the weight ratio between the
oxygen and hydrogen in water is eight to one, the atomic
weight of oxygen must be sixteen times that of hydrogen.

Through such considerations, supported by certain
other rules, it has gradually proved possible to obtain
reliable figures for the ratios between the atomic weights of
all known elements and the atomic weight of hydrogen.
For convenience it was customary to assign the number I
to the latter and to call the ratio between the weight of
the atom of a given element and the weight of the
hydrogen atom the atomic weight of the element in
question. Thus the atomic weight of oxygen is 16,
that of carbon 12, because one carbon atom weighs as
much as 12 hydrogen atoms. Nitrogen has the atomic
weight 14, sulphur 32, copper 63-6, etc.

14 THE ATOM AND THE BOHR THEORY

A summary of the chemical properties and chemical
compounds was greatly facilitated by the symbolic
system initiated by the Swedish chemist, Berzelius. In
this system the initial of the Latin name of the element
(sometimes with one other letter from the Latin name)
is made to indicate the element itself, an atom of the
element, and its atomic weight with respect to hydrogen
as unity, while a small subscript to the initial designates
the number of atoms to be used. For example, in the
chemical formula for sulphuric acid, H2S04, the symbolic
formula means that this substance is a chemical com¬
pound of hydrogen, sulphur and oxygen, that the acid
molecule consists of two atoms of hydrogen, one atom of
sulphur and four atoms of oxygen, and that the weight
ratios between the three constituent parts is as 2x1=2
to 32 to 4x16=64, or as 1 : 16 : 32. To say that the
chemical formula of zinc chloride is ZnCl2 means that
the zinc chloride molecule consists of one atom of zinc
and two atoms of chlorine. Furthermore the changes
which take place in a chemical process may be indi¬
cated in a very simple way. Thus the decomposition
of water into hydrogen and oxygen may be represented
by the so-called chemical “ equation ” 2H2CH*2H2-f 02,
where H2 and 02 signify the molecules of hydrogen and
oxygen respectively. Conversely, the combination of
hydrogen and oxygen to form water will be given by the
equation 2H2 + 02->2H20.

As a consequence of the development of the atomic
theory the atoms of the elements became, so to speak, the
building stones of which the elements and the chemical
compounds are built. It can also be said that the atoms
are the smallest particles which the chemists reckon with
in the chemical processes, but it does not follow from the

ATOMS AND MOLECULES

theory that these building stones in themselves are in¬
divisible. The theory leaves the way open to the idea
that they are composed of smaller parts. A belief
founded on such an idea was indeed enunciated by the
Englishman, Prout, a short time after Dalton had
developed his atomic theory. Prout assumed that
the hydrogen atoms were the fundamental ones, and
that the atoms of the other elements consisted of a
smaller or larger number of the atoms of hydrogen.
This might explain the fact that within the limits of
experimental error, many atomic weights seemed to
be integral multiples of that of hydrogen — 16 for
oxygen, 14 for nitrogen, and 12 for carbon, etc. This
led to the possibility that the same might hold for all
elements, and this hypothesis gave impetus to very
careful determinations of atomic weights. These, how¬
ever, showed that the assumption of the integral
multiples could not be verified. It therefore seemed
as if Prout’s hypothesis would have to be given up. It
has, however, recently come into its own again, al¬
though the situation is more complicated than Prout
had imagined (see p. 97).

Dalton’s atomic theory gave no information about
the atoms except that the atoms of each element had
a definite constant weight, and that they could combine
to form molecules in certain simple ratios. What the
forces are which unite them into such combinations,
and why they prefer certain unions to others, were
very perplexing problems, which could only be solved
when chemical and physical research had collected a
great mass of information as a surer source of speculation.

From the knowledge of atomic weights it was easy
to calculate what weight ratios might be found to exist

16 THE ATOM AND THE BOHR THEORY

in chemical compounds, the molecules of which con¬
sisted of simple atomic combinations. Thus many
compounds which were later produced in the laboratory
were first predicted theoretically, but only a small part
of the total number of possible compounds (corre¬
sponding to simple atomic combinations) could actually
be produced. Clearly it was one of the greatest problems
of chemistry to find the laws governing these cases.

It had early been known that the elements seemed
to fall into two groups, characterized by certain funda¬
mental differences, the metals and the metalloids.
In addition, there were recognized two very important
groups of chemical compounds, i.e. acids and bases,
possessing the property of neutralizing each other to
form a third group of compounds, the so-called salts.
The phenomenon called electrolysis, in which an electric
current separates a dissolved salt or an acid into two
parts which are carried respectively with and against
the direction of the current, indicates strongly that the
forces holding the atoms together in the molecule are of
an electrical nature, i.e. of the same nature as those
forces which bring together bodies of opposite electrical
charges. One is led to denote all metals as electro¬
positive and all metalloids as electronegative, which
means that in a compound consisting of a metal and a
metalloid the metal appears with a positive charge and
the metalloid with a negative charge. The chemist
Berzelius did a great deal to develop electrical theories
for chemical processes. Great difficulties, however,
were encountered, some proving for the time being in¬
surmountable. Such a difficulty, for example, is the
circumstance that two atoms of the same kind (like two
hydrogen atoms) can unite into a diatomic molecule,

ATOMS AND MOLECULES

although one might expect them to be similarly electrified
and to repel rather than attract each other.

Another circumstance playing a very important
part in determining the chemical compounds which are
possible, is the consideration of what is called valence .

As mentioned above, one atom of oxygen combines
with two atoms of hydrogen to form water, while one
atom of chlorine combines with but one atom of hydro¬
gen to form hydrogen chloride. The oxygen atom thus
seems to be “ equivalent ” to two hydrogen atoms or two
chlorine atoms, while one chlorine atom is “ equivalent ”
to one hydrogen atom. The atoms of hydrogen and
chlorine are for this reason called monovalent, while that
of oxygen is called divalent. Again an acid is a chemical
compound containing hydrogen, in which the hydrogen
can be replaced by a metal to produce a metallic salt.
Thus, when zinc is dissolved by sulphuric acid to form
hydrogen and the salt zinc sulphate, the hydrogen of
the acid is replaced by the zinc and the chemical change
may be expressed by the formula

Zn + H2S04-> H2 + ZnS04

In this, one atom of zinc changes place with two atoms
of hydrogen. The zinc atom is therefore divalent.
This is consistent with the fact that one zinc atom will
combine with one oxygen atom to form zinc oxide. To
take another example, if silver is dissolved in nitric
acid, one atom of silver is exchanged for one atom of
hydrogen. Silver, therefore, is monovalent, and we
should expect that one atom of oxygen would unite with
two atoms of silver. Some elements are trivalent, as,
for example, nitrogen, which combines with hydrogen to

form ammonia, NH3 ; others, again, are tetra valent, such
2

18 THE ATOM AND THE BOHR THEORY

as carbon, which unites with hydrogen to form marsh
gas CH4, and with oxygen to form carbon dioxide C02.
A valence greater than seven or eight has not been found
in any element.

A B

Fig. 3. — -Rough illustrations of the valences of the elements.

A. Hydrogen chloride (HC1) ; B. Water (HtO) ; C. Methane (CH4) ;

D. Ethylene (C2H4).

If we consider the matter rather roughly and more
or less as Gassendi did, we can explain the concept of
valence by assuming that the atoms possess hooks ;
thus hydrogen and chlorine are each furnished with
one hook, oxygen and zinc with two hooks, nitrogen with

ATOMS AND MOLECULES

three hooks, etc. When a hydrogen atom and a chlorine
atom are hooked together, there are no free hooks left,
and consequently the compound is said to be saturated.
When one hydrogen atom is hooked into each of the hooks
of an oxygen or carbon atom the saturation is also
complete (see Fig. 3, A, B, C).

The matter is not so simple as this, however, since
the same element can often appear with different valences.
Iron may be divalent, tri valent or hexa valent in different
compounds. In many cases, however, where an examina¬
tion of the weight ratios seems to show that an element
has changed its valence, this is not really true. It was
mentioned previously that carbon forms another com¬
pound with hydrogen in addition to CH4, namely, ethy¬
lene, containing half as much hydrogen in proportion
to the same amount of carbon. With the aid of Avo-
gadro’s Law, it is found that the ethylene molecule is
not CH2 but C2H4. Thus we can say that the two carbon
atoms in the molecule are held together by two pairs of

hooks, and consequently the compound can be expressed
_ . . . H— C— H

by the formula \\ where the dashes correspond to

H— C— H

hooks (cf. Fig. 3, D). Such a formula is called a
structural formula.

Even if we are not allowed to think of the atoms in
the molecules as held together by hooks, it is well to have
some sort of concrete picture of molecular structure.
It is possible to represent the tetravalent carbon atom
in the form of a tetrahedron, and to consider the united
atoms or atomic groups as placed at the four vertices.
With such a spatial representation we can get an idea
about many chemical questions which otherwise would
be difficult to explain. We know, for example, that

THE ATOM AND THE BOHR THEORY

two compound molecules having the same kind and
number of atoms and the same bonds (and hence the
same structural formulae), may yet be different in that
they are images of each other like a pair of gloves. Sub¬
stances whose molecules are symmetrical in this way
can be distinguished from each other by their different
action on the passage of light. This molecular chemistry
of space, or stereo-chemistry as it is called, has proved
of great importance in explaining difficult problems in
organic chemistry, i.e. the chemistry of carbon. Al¬
though there have never been many chemists who
really have believed the carbon atom to be a rigid
tetrahedron, we must admit that in this way it has
been possible to get on the track of the secrets of
atomic structure.

In comparing the properties of the elements with
their atomic weights, there has been discovered a
peculiar relation which remained for a long time without
explanation, but which later suggested a certain connec¬
tion between the inner structure of an atom and its
chemical properties. We refer to the natural or periodic
system of the elements which was enunciated in 1869 by
the Russian chemist, Mendelejeff, and about the same
time and independently by the German, Lothar Meyer.
This system will be understood most clearly by examining
the table on p. 23, where the elements with their respec¬
tive atomic weights and chemical symbols are arranged
in numbered columns so that the atomic weights increase
upon reading the table from left to right or from top to
bottom. It will be seen that in each of the nine columns
there are collected elements with related properties,
forming what may be called chemical families. The
table as here given is of a recent date and differs from

ATOMS AND MOLECULES

the old table of Mendelejeff, both in the greater number
of elements and in the particulars of the arrangement.
With each element there is associated a number which
indicates its position in the series with respect to
increasing atomic weight. Thus hydrogen has the num¬
ber i, helium 2, etc., up to uranium, the atom of which
is the heaviest of any known element, and to which the
number 92 is given. In each of the columns the elements
fall naturally into two sub-groups, and this division is
indicated in the table by placing the chemical symbols
to the right or left in the column.

On close examination it becomes evident that the
regularity in the system is not entirely simple. First
of all some cases will be found where the atomic weight
of one element is greater than that of the following
element. (The cases of argon and potassium on the one
hand and cobalt and nickel on the other are examples.)
Such an interchange is absolutely necessary if the
elements which belong to the same chemical family are
to be placed in the same column. As a second instance
of irregularity, attention must be called to Column VIII.
in the table. While in the first score or so of elements
it is always found that two successive elements have
different properties and clearly belong to distinct
chemical families, in the so-called iron group (iron,
cobalt and nickel) we meet with a case where success¬
ive elements resemble each other in many respects
(for instance, in their magnetic properties). Since there
are two more such “ triads ” in the periodic system,
however, we cannot properly call this an irregularity.
But in addition to these difficulties there is what we
may even call a kind of inelegance presented by the so-
called “ rare earths ” group. In this group there follow

22 THE ATOM AND THE BOHR THEORY

after lanthanum thirteen elements whose properties are
rather similar, so that it is very difficult to separate
them from each other in the mixtures in which they
occur in the minerals of nature. (In the table these
elements are enclosed in a frame.)

On the other hand, the apparent absence of an
element in certain places in the table (indicated by a
dash) cannot by any means be looked upon as irregular.
In Mendelejeff’s first system there were many vacant
spaces. With the help of his table Mendelejeff was, to
some extent, able to predict the properties of the missing
elements. An example of this is the case of the element
between gallium and arsenic. This is called germanium,
and was discovered to have precisely the properties
which had been predicted for it — a discovery which was
one of the greatest triumphs in favour of the reality
of the periodic system. On the whole, the elements
discovered since the time of Mendelejeff have found
their natural positions in the table. This is seen, for
example, in the case of the so-called “inactive gases”
of the atmosphere, helium, argon, neon, xenon, krypton
and niton, which have the common property of being
able to form no chemical combinations whatever. Their
valence is therefore zero, and in the table they are
placed by themselves in a separate column headed
with zero.

To explain the mystery of the periodic system, it was
necessary to make clear not only the regularity of it,
but also the apparent irregularities which seemed to be
arbitrary individual peculiarities of certain elements or
groups. In the periodic system, chemistry laid down
some rather searching tests for future theories of atomic
structure.

THE PERIODIC OR NATURAL SYSTEM OF THE ELEMENTS

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Ru ioi"7 Rh 102*9

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Sm 150*4

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Dy 162*5

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d Cp 175*0

76 Osmium 77 lridiun

Os 190*9 Ir 192*

78 Platinum

Pt 195*2

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Tm t68’5 Yb 173*1

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O 16

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C 12*0

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Si 28*3

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Ca 40*1

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Sr 87*6

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Ba 137*3

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24 THE ATOM AND THE BOHR THEORY

The Molecular Theory of Physics.

From a consideration of the chemical properties of
the elements we shall now turn to an examination of
the physical characteristics, although in a certain sense
chemistry itself is but one special phase of physics.

If matter is really constructed of independently
existing particles — atoms and molecules — the interplay
of the individual parts must determine not only the
chemical activities, but also the other properties of
matter. Since most of these properties are different
for different substances, or in other words are “ molecular
properties/' it is reasonable to suppose that in many
cases explanations can be more readily given by con¬
sidering the molecules as the fundamental parts. It is
natural that the first attempts to develop a molecular
theory concerned gases, for their physical properties are
much simpler than those of liquids or solids. This
simplicity is indeed easily understood on the molecular
theory. When a liquid by evaporation is transformed
into a gas, the same weight of the element has a vol¬
ume several hundred times greater than before. The
molecules, packed together tightly in the liquid, in the
gas are separated from each other and can move freely
without influencing each other appreciably. When two
of them come very close to each other, mutually re¬
pulsive forces will arise to prevent collision. Since it
must be assumed that in such a “ collision ” the individual
molecules do not change, they can then to a certain extent
be considered as elastic bodies, spheres for instance.

From considerations of this nature the kinetic theory
of gases developed. According to this a mass of gas
consists of an immense number of very small molecules.

ATOMS AND MOLECULES

Each molecule travels with great velocity in a straight
line until it meets an obstruction, such as another
molecule or the wall of the containing vessel ; after
such an encounter the molecule travels in a second
direction until it collides again, and so on. The
pressure of the gas on the wall of the container is the
result of the very many collisions which each little piece
of wall receives in a short interval of time. The magni¬
tude of the pressure depends upon the number, mass
and velocity of the molecules. The velocity will be
different for the individual molecules in a gas, even if all
the molecules are of the same kind, but at a given
temperature an average velocity can be determined and
used. If the temperature is increased, this average
molecular velocity will be increased, and if at the same
time the volume is kept constant, the pressure of the
gas on the walls will be increased. If the tempera¬
ture and the average velocity remain constant while
the volume is halved, there will be twice as many mole¬
cules per cubic centimetre as before. Therefore, on
each square centimetre of the containing wall there will
be twice as many collisions, and consequently the
pressure will be doubled. Boyle’s Law, that the pressure
of a gas at a given temperature is inversely proportional
to its volume, is thus an immediate result of the molecular
theory.

The molecular theory also throws new light upon
the correspondence between heat and mechanical work
and upon the law of the conservation of energy, which
about the middle of the nineteenth century was enunci¬
ated by the Englishman, Joule, the Germans, Mayer
and Helmholtz, and the Dane, Colding. A brief dis¬
cussion of heat and energy will be given here, since

26 THE ATOM AND THE BOHR THEORY

some conception of these phenomena is necessary in
understanding what follows.

To lift a stone of 5 pounds through a distance of
10 feet demands an expenditure of work amounting to
5x10=50 foot-pounds; but the stone is now enabled
to perform an equally large amount of work in falling
back these 10 feet. The stone, by its height above the
earth and by the attraction of the earth, now has in its
elevated position what is called “ potential ” energy to
the amount of 50 foot-pounds. If the stone as it falls
lifts another weight by some such device as a block and
tackle, the potential energy lost by the falling stone will
be transferred to the lifted one. If the apparatus is
frictionless, the falling stone can lift 5 pounds 10 feet or
10 pounds 5 feet, etc., so that all the 5° foot-pounds of
potential energy will be stored in the second stone. If
instead of being used to lift the second stone, the original
stone is allowed to fall freely or to roll down an inclined
plane without friction, the velocity will increase as the
stone falls, and, as the potential energy is lost, another
form of energy, known as energy of motion or kinetic
energy , is gained. Conversely, a body when it loses its
velocity can do work, such as stretching a spring or setting
another body in motion. Let us suppose that the stone
is fastened to a cord and is swinging like a pendulum
in a vacuum where there is no resistance to its motion.
The pendulum will alternately sink and rise again to
the same height. As the pendulum sinks, the potential
energy will be changed into kinetic energy, but as it
rises again the kinetic will be exchanged for potential.
Thus there is no loss of energy, but merely a continuous
exchange between the two forms.

If a moving body meets resistance, or if its free fall is

ATOMS AND MOLECULES

halted by a fixed body, it might seem as if, at last, the
energy were lost. This, however, is not the case, for
another transformation occurs. Every one knows that
heat is developed by friction, and that heat can produce
work, as in a steam-engine. Careful investigations have
shown that a given amount of mechanical work will
always produce a certain definite amount of heat, that is,
400 foot-pounds of work, if converted into heat, will
always produce 1 B.T.U. of heat, which is the amount
necessary to raise the temperature of 1 pound of water
i° F. Conversely, when heat is converted into work,
1 B.T.U. of heat “ vanishes ” every time 400 foot-pounds
of work are produced. Heat then is just a special form
of energy, and the development of heat by friction or
collision is merely a transformation of energy from one
form to another.

With the assistance of the molecular theory it becomes
possible to interpret as purely mechanical the transforma¬
tion of mechanical work into heat energy. Let us suppose
that a falling body strikes a piston at the top of a gas-
filled cylinder, closed at the bottom. If the piston is
driven down, the gas will be compressed and therefore
heated, for the speed of the molecules will be increased
by collisions with the piston in its downward motion.
In this example the kinetic energy given to the piston by
the exterior falling body is used to increase the kinetic
energy of the molecules of the gas. When the molecules
contain more than one atom, attention must also be
given to the rotations of the atoms in a molecule about
each other. A part of any added kinetic energy in the
gas will be used to increase the energy of the atomic
rotations.

The next step is to assume that, in solids and liquids,

28 THE ATOM AND THE BOHR THEORY

heat is purely a molecular motion. Here, too, the
development of heat after collision with a moving body
should be treated as a transformation of the kinetic
energy of an individual, visible body into an inner kinetic
energy, divided among the innumerable invisible mole¬
cules of the heated solid or liquid. In considering the
internal conduct of gases it is unnecessary (at least in
the main) to consider any inner forces except the re¬
pulsions in the collisions of the molecules. In solids
and liquids, however, the attractions of the tightly
packed molecules for each other must not be neglected.
Indeed the situation is too complicated to be explained
by any simple molecular theory. Not all energy trans¬
formations can be considered as purely mechanical.
For instance, heat can be produced in a body by rays
from the sun or from a hot fire, and, conversely, a hot
body can lose its heat by radiation. Here, also, we
are concerned with transformations of energy ; therefore
the law for the conservation of energy still holds, i.e. the
total amount of energy can neither be increased nor
decreased by transformations from one form to another.
For the production of i B.T.U. of heat a definite amount
of radiation energy is required; conversely, the same
amount of radiation energy is produced when i B.T.U.
of heat is transformed into radiation. This change
cannot, however, be explained as the result of mechanical
interplay between bodies in motion.

The mechanical theory of heat is very useful when
we restrict ourselves to the transfer of heat from one
body to another, which is in contact with it. When
applied to gases the theory leads directly to Avogadro’s
Law. If two masses of gas have the same tempera¬
ture, i.e., if no exchange of heat between them takes

ATOMS AND MOLECULES

place even if they are in contact with each other, then
the average value of the kinetic energy of the molecules
must be the same in both gases. If one gas is hydrogen
and the other oxygen, the lighter hydrogen molecules
must have a greater velocity than the heavier oxygen
molecules ; otherwise they cannot have the same
kinetic energy (the kinetic energy of a body is one-half
the product of the mass and the square of the velocity).
Since the pressure of a gas depends upon the kinetic
energy of the molecules and upon their number per cubic
centimetre, at the same temperature and pressure equal
volumes must contain equal numbers of oxygen and of
hydrogen molecules. As Joule showed in 1851, from
the mass of a gas per cubic centimetre and from its
pressure per square centimetre, the average velocity
of the molecules can be calculated. For hydrogen at
o° C. and atmospheric pressure the average velocity
is about 5500 feet per second ; for oxygen under the
same conditions it is something over 1300 feet per
second.

All these results of the atomic and molecular theory,
however, gave no information about the absolute weight
of the individual atoms and molecules, nor about their
magnitude nor the number of molecules in a cubic centi¬
metre at a given temperature and pressure. As long as
such questions were unsolved there was a suggestion of
unreality in the theory. The suspicion was easily aroused
that the theory was merely a convenient scheme for pic¬
turing a series of observations, and that atoms and mole¬
cules were merely creations of the imagination. The
theory would seem more plausible if its supporters could
say how large and how heavy the atoms and molecules
were. The molecular theory of gases showed how to

30 THE ATOM AND THE BOHR THEORY

solve these problems which chemistry had been powerless
to solve.

Let ns assume that the temperature of a mass of
gas is ioo° C. at a certain altitude, and o° C. one metre
lower, i.e., the molecules have different average velocities
in the two places. The difference between the velocities
will gradually decrease and disappear on account of
molecular collisions. We might expect this “ levelling
out ” process or equilibration to proceed very rapidly
because of the great velocity of the molecules, but we
must consider the fact that the molecules are not
entirely free in their movements. In reality they will
travel but very short distances before meeting other
molecules, and consequently their directions of motion
will change. It is easy to understand that the differ¬
ence between the velocities of the molecules of the
gas will not disappear so quickly when the molecules
move in zigzag lines with very short straight stretches.
The greater velocity in one part of the gas will then
influence the velocity in the other part only through
many intermediate steps. Gases are therefore poor
conductors of heat. When the molecular velocity of a
gas and its conductivity of heat are known, the average
length of the small straight pieces of the zigzag lines
can be calculated— in other words, the length of the
mean free path. This length is very short ; for oxygen
at standard temperature and pressure it is about one
ten-thousandth of a millimetre, or o*i [i>} where (M is
0*001 millimetre or one micron.

In addition to the velocity of the molecules, the
length of the mean free path depends upon the average
distance between the centres of two neighbouring mole¬
cules (in other words, upon the number of molecules

ATOMS AND MOLECULES

per cubic centimetre) and upon their size. There is
difficulty in defining the size of molecules because, as a
rule, each contains at least two atoms ; but it is help¬
ful to consider the molecules, temporarily, as elastic
spheres. Even with this assumption we cannot yet
determine their dimensions from the mean free path,
since there are two unknowns, the dimensions of the
molecules and their number per cubic centimetre. Upon
these two quantities depends, however, also the volume
which will contain this number of molecules, if they are
packed closely together. If we assume that we meet
such a packing when the substance is condensed in
liquid form, this volume can be calculated from a
knowledge of the ratio between the volume in liquid
form and the volume of the same mass in gaseous form
(at o° C. and atmospheric pressure). Then from this
result and the length of the mean free path the two
unknowns can be determined. Although the assump¬
tions are imperfect, they serve to give an idea about
the dimensions of the molecules ; the results found in
this way are of the same order of magnitude as those
derived later by more perfect methods of an electrical
nature.

The radius of a molecule, considered as a sphere,
is of the order of magnitude o-i where ^ means
io-6 millimetre or o-ooi micron. Even if a mole¬
cule is by no means a rigid sphere, the value given
shows that the molecule is almost unbelievably small,
or, in other words, that it can produce appreciable
attraction and repulsion in only a very small region
in space.

The number of molecules in a cubic centimetre of
gas at o° C. and atmospheric pressure has been calculated

THE ATOM AND THE BOHR THEORY

with fair accuracy as approximately 27 X io18. From
this number and from the weight of a cubic centimetre
of a given gas the weight of one molecule can be found.
One hydrogen molecule weighs about 1-65 Xio-24 grams,
and one gram of hydrogen contains about 6xio23
atoms and 3 X io23 molecules. The weight of the atoms
of the other elements can be found by multiplying the
weight of the hydrogen atom by the relative atomic
weight of the element in question — 16 for oxygen, 14
for nitrogen, etc. If the pressure on the gas is reduced
as much as possible (to about one ten-millionth of an
atmosphere) there will still be 3X1012 molecules in a
cubic centimetre, and the average distance between
molecules will be about one micron. The mean free path
between two collisions will be considerable, about two
metres, for instance, in the case of hydrogen.

The values found for the number, weight and
dimensions of molecules are either so very large or so
extremely small that many people, instead of having
more faith in the atomic and molecular theory, perhaps
may be more than ever inclined to suppose the atoms
and molecules to be mere creations of the imagination.
In fact, it is only two or three decades ago that some
physicists and chemists — led by the celebrated German
scientist, Wilhelm Ostwald— denied the existence of
atoms and molecules, and even went so far as to try to
remove the atomic theory from science. When these
sceptics, in defence of their views, said that the atoms
and molecules were, and for ever would be, completely
inaccessible to observation, it had to be admitted at
that time that they were seemingly sure of their argu¬
ment, in this one objection at any rate.

A series of remarkable discoveries at the close of

ATOMS AND MOLECULES

the nineteenth century so increased our knowledge of
the atoms and improved the methods of studying
them that all doubts about their existence had to be
silenced. However incredible it may sound, we are
now in a position to examine many of the activities of a
single atom, and even to count atoms, one by one,
and to photograph the path of an individual atom. All
these discoveries depend upon the behaviour of atoms as
electrically charged, moving under the influence of elec¬
trical forces. This subject will be developed in another
section after a discussion of some phenomena of light,
an understanding of which is necessary for the apprecia¬
tion of the theory of atomic structure proposed bv Niels
Bohr.

In the molecular theory of gases, where we have to
do with neutral molecules, much progress has in the last
years been made by the Dane, Martin Knudsen, in his
experiments at a very low pressure, when the molecules
can travel relatively far without colliding with other
molecules. While his researches give information on
many interesting and important details, his work gives
at the same time evidence of a very direct nature con¬
cerning the existence of atoms and molecules.

CHAPTER II

LIGHT WAVES AND THE SPECTRUM

The Wave Theory of Light.

There have been several theories about the nature
of light. The great English physicist, Isaac Newton
(1642-1727), who was a pioneer in the study of light as
well as in that of mechanics, favoured an atomic ex¬
planation of light ; i.e.y he thought that it consisted of
particles or light corpuscules which were emitted from
luminous bodies like projectiles from a cannon. In
contrast to this “ emission ” theory was the wave theory
of Newton’s contemporary, the Dutch scientist, Huygens.
According to him, light was a wave motion passing from
luminous bodies into a substance called the ether, which
filled the otherwise empty universe and permeated all
bodies, at least all transparent ones. In the nineteenth
century the wave theory, particularly through the work
of the Englishman, Young, and the Frenchman, Fresnel,
came to prevail over the emission theory. Since the
wave theory plays an important part in the following
chapters, a discussion of the general characteristics
of all wave motions is appropriate here. The examples
will include water waves on the surface of a body of
water, and sound waves in air.

Let us suppose that we are in a boat which is anchored

LIGHT WAVES AND THE SPECTRUM

on a body of water and let us watch the regular waves
which pass us. If there is neither wind nor current, a
light body like a cork, lying on the surface, rises with
the wave crests and sinks with the troughs, going forward
slightly with the former and backward with the latter,
but remaining, on the whole, in the same spot. Since
the cork follows the surrounding water particles, it shows
their movements, and we thus see that the individual

Fig. 4. — Photograph of the interference Fig. 5. — A section of the same picture

between two similar wave systems. enlarged.

(From Grimsehl, Lehrbuch dev Physik.)

particles are in oscillation, or more accurately, in circu¬
lation, one circulation being completed during the time
in which the wave motion advances a wave-length, i.e., the
distance from one crest to the next. This interval of
time is called the time of oscillation , or the period. If the
number of crests passed in a given time is counted, the
oscillations of the individual particles in the same time
can be determined. The number of oscillations in the
unit of time, which we here may take to be one minute,
is called the frequency. If the frequency is forty and the

THE ATOM AND THE BOHR THEORY

wave-length is three metres, the wave progresses 3X40
= 120 metres in one minute. The velocity with which
the wave motion advances, or in other words its velocity
of propagation , is then 120 metres per minute. We thus
have the rule that velocity of propagation is equal to the
product of frequency and wave-length (cf. Fig. 8).

On the surface of a body of water there may exist
at the same time several wave systems ; large waves
created by winds which have themselves perhaps died
down, small ripples produced by breezes and running over
the larger waves, and waves from ships, etc. The form
of the surface and the changes of form may thus be very
complicated ; but the problem is simplified by com¬
bining the motions of the individual wave systems at
any given point. If one system at a given time gives a
crest and another at the same instant also gives a crest
at the same point, the two together produce a higher
crest. Similarly, the resultant of two simultaneous
troughs is a deeper trough ; a crest from one system and
a simultaneous trough from the other partially destroy
or neutralize each other. A very interesting yet simple
case of such “ interference ”• of two wave systems is
obtained when the systems have equal wave-lengths and
-equal amplitudes. Such an interference can be pro¬
duced by throwing two stones, as much alike as possible,
into the water at the same time, at a short distance from
each other. When the two sets of wave rings meet
' there is created a network of crests and troughs.
Figs. 4 and 5 show photographs of such an interference,
produced by setting in oscillation two spheres which
were suspended over a body of water.

In Fig. 6 there is a schematic representation of an
interference of the same nature. Let us examine the

LIGHT WAVES AND THE SPECTRUM

situation at points along the lower boundary line. At
o, which is equidistant from the two wave centres, there
is evidently a wave crest in each system ; therefore
there is a resultant crest of double the amplitude of a
single crest if the two systems have the same amplitude.
Half a period later there is a trough in each system with

Fig. 6. — Schematic representation of an interference.

a resultant trough of twice the amplitude of a single
trough. Thus higher crests and deeper troughs alter¬
nate. The same situation is found at point 2, a wave¬
length farther from the left than from the right wave
centre ; in fact, these results are found at all points
such as 2, 2', 4 and 4', where the difference in distance

THE ATOM AND THE BOHR THEORY

from the two wave centres is an even number of wave¬
lengths. At the point i, on the other hand, where the
difference between the distance from the centres is one-
half a wave-length, a crest from one system meets a
trough from the other, and the resultant is neither crest
nor trough but zero. There is the same result at points
i', 3, 3', 5, 5', etc., where the difference between the
distances from the two wave centres is an odd number
of half wave-lengths. By throwing a stone into the
water in front of a smooth wall an interference is ob¬
tained, similar to the one described above. The waves

Fig. 7. — Waves which are reflected by a board and pass

through a hole in it.

are reflected from the wall as if they came from a centre
at a point behind the wall and symmetrically placed
with respect to the point where the stone actually falls.

When a wave system meets a wall in which there is
a small hole, this opening acts as a new wave centre,
from which, on the other side of the wall, there spread
half-rings of crests and troughs. But if the waves are
small and the opening is large in proportion to the wave¬
length, the case is essentially different. Let us suppose
that wave rings originate at every point of the opening.
As a result of the co-operation of all these wave systems
the crests and troughs will advance, just as before, in
the original direction of propagation, i.e.f along straight

LIGHT WAVES AND THE SPECTRUM

lines drawn from the original wave centre through the
opening ; lines of radiation, we may call them. It can
be shown, however, that as these lines of radiation
deviate more and more from the normal to the wall, the
interference between wave systems weakens the resultant
wave motion. If the deviation from the normal to the
wall is increased, the weakening varies in magnitude,
provided that the waves are sufficiently small ; but even
if the wave motions at times may thus “ flare up ” some¬
what, still on the whole they will decrease as the deviation
from the normal to the wall is increased. The smaller
the waves in comparison to the opening, the more
marked is the decrease of the wave motions as the
distance from the normal to the wall is increased, and
the more nearly the waves will move on in straight lines.
That light moves in straight lines, so that opaque
objects cast sharp shadows, is therefore consistent with
the wave theory, provided the light waves are very
small ; though it is reasonable to expect that on the
passage of light through narrow openings there will
be produced an appreciable bending in the direction
of the rays. This supposition agrees entirely with
experiment. As early as the middle of the seventeenth
century, the Italian Grimaldi discovered such a dif¬
fraction of light which passes through a narrow opening
into a dark room.

In both light and sound the use of such terms as
wave and wave motion is figurative, for crests and
troughs are lacking. But this choice of terms is com¬
mendable, because sound and light possess an essential
property similar to one possessed by water waves.
What happens when a tuning-fork emits sound-waves
into the surrounding air, is that the air particles are set

40 THE ATOM AND THE BOHR THEORY

in oscillation in the direction of the propagation of sound.
All the particles of air have the same period as the tuning-
fork, and the number of oscillations per second deter¬
mines the pitch of the note produced ; but the air
particles at different distances from the tuning-fork
are not all simultaneously in the same phase or condition
of oscillation. If one particle, at a certain distance
from the source of sound and at a given time, is moving
most rapidly away from the source, then at the same

Fig. 8. — Schematic representation of a wave.

A and B denote crests ; C denotes a trough.

X = wave-length, a = amplitude of wave.

If T denotes the time the wave takes to travel from A to B, and
v=i/T the frequency, the wave velocity v will be equal to X/T = Xi'.

Points P and P' are points in the same phase.

time there is another particle, somewhat farther along
the direction of propagation, which is moving towards
the source most rapidly. This alternation of direction
will exist all along the path of the sound. Where the
particles are approaching each other, the air is in a state
of condensation, and where the particles are drawing
apart, the air is in a state of rarefaction. While the
individual particles are oscillating in approximately the
same place, the condensations and rarefactions, like
troughs and crests in water, advance with a velocity
which is called the velocity of sound. If we call the

LIGHT WAVES AND THE SPECTRUM

distance between two consecutive points in the same
phase a wave-length, and the number of oscillations in a
period of time the frequency, then, as in the case of
water waves, the velocity of propagation will be equal
to the product of frequency and wave-length.

Light, like sound, is a periodic change of the condi¬
tions in the different points of space. These changes
which emanate from the source of light, in the course
of one period advance one wave-length, i.e., the distance
between two successive points in the same phase and
lying in the direction of propagation. As in the cases of
sound and water waves, the velocity of propagation or the
velocity of light is equal to the product of frequency and
wave-length. If this velocity is indicated by the letter
c, the frequency by v and the wave-length by a, then

c c

c=v\ or v= - or a =

A V

The velocity of light in free space is a constant, the
same for all wave-lengths. It was first determined by
the Danish astronomer Ole Rpmer (1676) by observa¬
tions of the moons of Jupiter. According to the measure¬
ments of the present day the velocity of light is about
1,000,000 feet or 300,000 kilometres per second. In
centimetres it is thus about 3 X io10.

Efforts have been made to consider light waves, like
sound waves, as produced by the oscillations of particles,
not of the air, but of a particular substance, the “ ether,”
filling and permeating everything ; but all attempts to
form definite representations of the material properties
of the ether and of the movements of its particles have
been unsuccessful. The electromagnetic theory of light ,
enunciated about fifty years ago by the Scottish physi-

42 THE ATOM AND THE BOHR THEORY

cist, Maxwell, has furnished information of an essentially
different character concerning the nature of light waves.

Let us suppose that electricity is oscillating in a
conductor connecting two metal spheres, for instance.
The spheres, therefore, have, alternately, positive and
negative charges. Then according to Maxwell’s theory
we shall expect that in the surrounding space there will
spread a kind of electromagnetic wave with a velocity equal
to that of light. Wherever these waves are, there should
arise electric and magnetic forces at right angles to each
other and to the direction of propagation of the waves ;
the forces should change direction in rhythm with the
movements of electricity in the emitting conductor.
By way of illustration let us assume that we have some¬
where in space an immensely small and light body or par¬
ticle with an electric charge. If, in the region in question,
an electromagnetic wave motion takes place, then the
charged particle will oscillate as a result of the periodically
changing electrical forces. The particle here plays the
same role as the cork on the surface of the water (cf . p. 35) ;
the charged body thus makes the electrical oscillations
in space apparent just as the cork shows the oscillations
of the water. In addition to the electrical forces there
are also magnetic forces in an electromagnetic wave.
We can imagine that they are made apparent by using a
very small steel magnet instead of the charged body.
According to Maxwell’s theory, the magnet exposed to
the electromagnetic wave will perform rapid oscillations.
Maxwell came to the conclusion that light consisted of
electromagnetic waves of a similar nature, but much
more delicate than could possibly be produced and made
visible directly by electrical means.

In the latter part of the nineteenth century the

LIGHT WAVES AND THE SPECTRUM

German physicist, H. Hertz, succeeded in producing
electromagnetic waves with oscillations of the order of
magnitude of 100,000,000 per second, corresponding to
wave-lengths of the order of magnitude of several metres.
(\=c/v= 3 x io10/io8=3OO cm.). Moreover, he proved the
existence of the oscillating electric forces by producing
electric sparks in a circle of wire held in the path of the
waves. He showed also that these electromagnetic waves
were reflected and interfered with each other according to
the same laws as in the case of light waves. After these
discoveries there could be no reasonable doubt that light
waves were actually electromagnetic waves, but so small
that it would be totally impossible to examine the oscilla¬
tions directly with the assistance of electric instruments.

But there was in this work of Hertz no solution of the
problems about the nature of ether and the processes
underlying the oscillations. More and more, scientists
have been inclined to rest satisfied with the electro¬
magnetic description of light waves and to give up
speculation on the nature of the ether. Indeed, within
the last few years, specially through the influence of
Einstein’s theory of relativity, many doubts have arisen
as to the actual existence of the ether. The disagree¬
ment about its existence is, however, more a matter of
definition than of reality. We can still talk about light
as consisting of ether waves if we abandon the old
conception of the ether as a rigid elastic body with
definite material properties, such as specific gravity,
hardness and elasticity.

The Dispersion of Light.

It has been said that the wave-length of light is
much shorter than that of the Hertzian waves. This

44 THE ATOM AND THE BOHR THEORY

does not mean that all light waves have the same wave¬
length and frequency. The light which comes to us
from the sun is composed of waves of many different
wave-lengths and frequencies, to each of which corre¬
sponds a particular colour.

In this respect also light may be compared with
sound. In whatever way a sound is produced, it is in
general of a complicated nature, composed of many
distinct notes, each with its characteristic wave-length
and frequency. Naturally the air particles cannot
oscillate in several different ways simultaneously. At
a given time, however, we can think of the condensa¬
tion and rarefactions of the air or the oscillations of the
particles corresponding to different tones, as com¬
pounded with each other in a way similar to that in
which the resultant crests and troughs are produced
on a body of water with several coexistent wave
systems. When we say that the complicated wave-
movement emitted from some sound-producing instru¬
ment consists of different tones, this does not only
mean that we may imagine it purely mathematically as
resolved into a series of simpler wave systems. The
resolution may also take place in a more physical
way. Let us assume that we have a collection of strings
each of which will produce a note of particular pitch.
Now, if sound waves meet this collection of strings,
each string is set in oscillation by the one wave in the
compound sound wave which corresponds to it. Each
string is then said to act as a resonator for the note in
question. The notes which set the resonator strings
in oscillation sound more loudly in the neighbourhood
of the resonators ; but, as the wave train continues on
its journey the tones taken out by the strings will become

LIGHT WAVES AND THE SPECTRUM

weak in contrast to those notes which found no corre¬
sponding strings. The resonator is said to absorb the
notes with which it is in pitch.

Light which is composed of different colours, i.e.,
of wave systems with different wave-lengths, can also
be resolved or dispersed, but by a method different
from that in the case of sound.

When light passes from one medium to another,
as from air to glass or vice versa, it is refracted, i.e., the
direction of the light rays is changed ; but if the light
is composed of different colours the refraction is accom¬
panied by a spreading ” of the colours which is called
dispersion. If we look through a glass prism so that
the light from the object examined must pass in and out
through two faces of the prism which make not too
great an angle with each other, the light-producing
object is not only displaced by the refraction, but has
coloured edges. Newton was the first to explain the
relation of the production of the colours to refraction.
He made an experiment with sunlight, which he sent
through a narrow opening into a dark room. The sun¬
light was then by a glass prism transformed or dis¬
persed into a band of colour, a spectrum, consisting of
all the colours of the rainbow, red, yellow, green, blue
and violet, in the order named, and with continuous
transition stages between neighbouring colours.

In Newton’s original experiment the different wave¬
lengths were but imperfectly separated. A spectrum with
pure wave-lengths can be obtained with a spectroscope
(cf. Fig. 9). The light to be investigated illuminates an
adjustable vertical slit in one end of a long tube, called the
collimator, with a lens in the other end. If the slit is in
the focal plane of the lens, the light at any point in the

46 THE ATOM AND THE BOHR THEORY

slit goes in parallel rays after meeting the lens. It then
meets a prism, with vertical edges, placed on a little re-
• volving platform. The rays, refracted by the prism, go

FIG. 9. — Prism spectroscope. To the right is seen the collimator,
to the left the telescope, in the foreground a scheme for
illuminating the cross- wire.

(From an old print.)

in a new direction into a telescope whose objective lens
gives in its focal plane, for every colour, a clear vertical
image of the slit. These images can be examined
through the ocular of the telescope ; but since the different
colours are not refracted equally, each coloured image

LIGHT WAVES AND THE SPECTRUM

of the slit has its own place. The totality of the slit
images then forms a horizontal spectrum of the same
height as the individual images. By revolving the
collimator different parts of the spectrum can be put
in the middle of the field of view. To facilitate measure¬
ments in the spectrum there is in the focal plane of
the collimator a sliding cross- wire with an adjusting
screw or a vertical strand of spider web.

A, grating; C, D, E . . . H, slits; M M, incident
rays. When D D' , EE'... are a whole number
of wave-lengths, the light waves which move in
the direction indicated by C N and are collected
by a lens, at the focal point will all be in the
same phase and therefore will reinforce each
other. In other directions the light action from
one slit is compensated by that from another.

Instead of using the refraction of light in a prism to
separate the wave-lengths, we can use the interference
which arises when a bundle of parallel light waves
passes through a ruled grating , consisting of a great
many very fine parallel lines, equidistant from each
other ; such a grating can be made by ruling lines with
a diamond point on the metal coating of a silvered
plate of glass. From each line there are sent out light

48 THEJATOM AND THE BOHR THEORY

waves in all directions ; but if we are considering light
of one definite colour (a given wave-length, mono¬
chromatic light), the interference among the waves from
all the slits practically destroys all waves except in the
direction of the original rays and in the directions
making certain angles with the former, dependent upon
the wave-length and the distance between two successive
lines (the grating space). Monochromatic light can be
obtained by using as the source of light a spirit flame,
coloured yellow with common salt (sodium chloride).
If the slit in a spectroscope is lighted with a yellow light
from such a flame, and if a grating normal to the direction
of the rays is substituted for the prism, then in the
telescope there is seen a yellow image of the slit, and on
each side of it one, two, three or more yellow images.
If sunlight is used the central image is white, since
all the colours are here assembled. The other images
become spectra because the different colours are un¬
equally refracted. In these grating spectra, which
according to their distance from the central fine are
called spectra of the first, second or third order, the
violet part lies nearest to the central line, the red part
farthest away. Since the deflection is the greater the
greater the wave-length, then violet light must have
the shortest wave-length and red the greatest. From
the amount of the refraction and the size of the grating
space the wave-length of the light under investigation
can be calculated.

For the yellow light from our spirit flame the wave¬
length is about 0*000589 mm. or 0*589 gu or 589 fjbfjj. In
centimetres the wave-length is 0*0000589 cm. ; from
the formula v=c/X, t'=526xio12. The frequency is
thus almost inconceivably large. For the most distant

LIGHT WAVES AND THE SPECTRUM

red and violet in the spectrum the wave-lengths are
respectively about 800 (&(/* and 400 {&(/,, and the frequencies
375 X io12 and 750 X io12 oscillations per second.

In scientific experiments a grating of specular metal
with parallel rulings is substituted for the transparent
grating. The spectrum is then given by the reflected
light from the parts between the rulings. Specular
gratings can be made by ruling on a concave mirror,
which focuses the rays so that a glass lens is unnecessary.
Gratings with several hundred lines or rulings to the
millimetre give excellent spectra, with strength of light
and marked dispersion. The preparation of the first
really good gratings is due to the experimental skill of
the American, Rowland, who in 1870 built a dividing
engine from which the greater part of the good gratings
now in use originate. The contribution which Rowland
thereby made to physical science can hardly be over¬
estimated.

Spectral Lines.

In the early part of the nineteenth century Wollas¬
ton, in England, and later Fraunhofer in Germany, dis¬
covered dark lines in the solar spectrum, a discovery
which meant that certain colours were missing. The
most noticeable of these so-called “ Fraunhofer Lines ”
were named with the letters A, B, C, D, E, F, G, H,
from red to violet. It was later discovered that some
of the lines were double, that the D-line, for instance,
can be resolved into Dj and D2 ; other letters, such as
b and h, were introduced to denote new lines. With
improvements in the methods of experiment and research
the number of lines has increased to hundreds and
even thousands. The light from a glowing solid or
4

50 THE ATOM AND THE BOHR THEORY

liquid element forms, on the other hand, a continuous
spectrum, i.e. a spectrum which has no dark lines.
An illustration of the solar spectrum with the strongest
Fraunhofer lines is given at the end of the book.

In contrast to the solar spectrum with dark lines on
a bright background are the so-called line spectra , which
consist of bright lines on a dark background. The first
known line spectrum was the one given by light from the
spirit flame coloured with common salt, mentioned in
connection with monochromatic light. As has been
said, this spectrum had just one yellow line which was
later found to consist of two lines close to each other.
It is sodium chloride which colours the flame yellow.
The colour is due, not to the chlorine, but to the sodium,
for the same double yellow line can be produced by
using other sodium salts not compounded with chlorine.
The yellow light is therefore called sodium light. No. 7
in the table of spectra at the end of the book shows the
spectrum produced by sodium vapour in a flame. (On
account of the small scale in the figure it is not shown
that the yellow line is double.)

Another interesting discovery was soon made,
namely, that the sodium fine has exactly the same wave¬
length as the light lacking in the solar spectrum, where
the double D-line is located. About i860 Kirchhoff
and Bunsen explained this remarkable coincidence as
well as others of the same nature. They showed by
direct experiment that if sodium vapour is at a high
temperature it can not only send out the yellow light, but
also absorb fight of the same wave-length when rays from
a still warmer glowing body pass through the vapour.
This phenomenon is something like that in the case of
sound waves where a resonator absorbs the pitch which

LIGHT WAVES AND THE SPECTRUM

it can emit itself. The existence of the dark D-line in
the solar spectrum must then mean that in the outer
layer of the sun there is sodium vapour present
of lower temperature than the white-hot interior
of the sun, and that the light corresponding to the
D-line is absorbed by the vapour. Several ingenious
experiments, which cannot be described here, have
given further evidence in favour of this explanation.

In the other line spectra, just as in that from the
common salt flame, definite lines correspond to definite
elements and not to chemical compounds. The
emission of these lines is then not a molecular char¬
acteristic, but an atomic one. The line spectra of metals
can often be produced by vaporizing a metallic salt in
a spirit flame or in a hot, colourless gas flame (from a
Bunsen burner). It is even better to use an electric arc
or strong electric sparks. The atoms from which
gaseous molecules are formed can also be made to emit
light which by means of the spectroscope is shown to
consist of a line spectrum. These results are obtained
by means of electric discharges of various kinds, arcs,
and spark discharges through tubes where the gas is in
a rarefied state.

The other Fraunhofer lines in the solar spectrum
correspond to bright lines in the fine spectra of certain
elements which exist here on earth. These Fraunhofer
lines must then be assumed to be caused by the absorp¬
tion of light by the elements in question. This may be
explained by the presence of these elements as gases in
the solar atmosphere, through which passes the light
from the inner layer. This inner surface would in itself
emit a continuous spectrum.

The work of Kirchhoff and Bunsen put at the disposal

THE ATOM AND THE BOHR THEORY

of science became a new tool of incalculable scope. First
and foremost, spectrum examinations were taken into the
service of chemistry as spectrum analysis. It has thus
become possible to analyse quantities of matter so small
that the general methods of chemistry would be quite
powerless to detect them. It is also possible by spectrum
analysis to detect minute traces of an element ; several
elements were in this way first discovered by the spectro¬
scope. Moreover, chemical analysis has been extended
to the study of the sun and stars. The spectral lines
have given us answers to many problems of physics —
problems which formerly seemed insoluble. Last but
not least spectrum analysis has given us a key to the
deepest secrets of the atom, a key which Niels Bohr
has taught us how to use.

In the discussion of the spectrum we have hitherto
restricted ourselves to the visible spectrum limited on
the one side by red and on the other by violet. But
these boundaries are in reality fortuitous, determined
by the human eye. The spectrum can be studied by
other methods than those of direct observation. The
more indirect methods include the effect of the rays on
photographic plates and their heating effect on fine
conducting wires for electricity, held in various parts of
the spectrum. It has thus been discovered that beyond
the visible violet end of the spectrum there is an ultra¬
violet region with strong photographic activity and an
infra-red region producing marked heat effects. There
are both dark and light spectral lines in these new parts
of the spectrum. The fact that glass is not transparent
to ultra-violet or infra-red rays has been an obstacle in
the experiments, but the difficulty can be overcome
by using other substances, such as quartz or rock salt,

LIGHT WAVES AND THE SPECTRUM

for the prisms and lenses, or by substituting concave
gratings. By special means it has been possible to
detect rays with wave-lengths as great as 300 (Jb and as
small as about 0-02 fjb, corresponding to frequencies
between io12, and 15 X io15 vibrations per second, while
the wave-lengths of the luminous rays lie between o-8
and 0*4 (j> . The term “ light wave ” is often used to

Fig. 1 1 . — Photographic effect of X-rays, which are passed through
the atom grating in a magnesia crystal.

refer to the ultra-violet and infra-red rays which can be
shown in the spectra produced by prisms or gratings.

The electrically produced electromagnetic waves, as
already mentioned, have wave-lengths much greater
than 300 fjj. In wireless telegraphy there are generally
used wave-lengths of one kilometre or more, correspond¬
ing to frequencies of 300,000 vibrations per second or
less. By direct electrical methods it has, however, not
been possible to obtain wave-lengths less than about one-

THE ATOM AND THE BOHR THEORY

half a centimetre, a length differing considerably from
the o*3 millimetre wave of the longest infra-red rays.
Wave-lengths much less than 0-02 or 20 fjbfjj exist in the
so-called Rontgen rays or X-rays with wave-lengths
as small as o-oi pf* corresponding to a frequency of
30X1018. These rays cannot possibly be studied even
with the finest artificially made gratings, but crystals,
on account of the regular arrangement of the atoms,
give a kind of natural grating of extraordinary fineness.
With the use of crystal gratings success has been at¬
tained in decomposing the Rongten rays into a kind of
spectrum, in measuring the wave-lengths of the X-rays
and in studying the interior structure of the crystals.
The German Laue, the discoverer of the peculiar action
of crystals on X-rays (1912), let the X-rays beams pass
through the crystal, obtaining thereby photographs of
the kind illustrated in Fig. 11. Later on essential
progress was due to the Englishmen, W. H. and W. L.
Bragg, who worked out a method of investigation by
which beams of X-rays are reflected from crystal faces.
The greatest wave-length which it has been possible to
measure for X-rays is about 1*5 which is still a long
way from the 20 {h[jj of the furthermost ultra-violet rays.

It may be said that the spectrum since Fraunhofer
has been made not only longer but also finer, for the
accuracy of measuring wave-lengths has been much
increased. It is now possible to determine the wave¬
length of a line in the spectrum to about o-ooi ftp or
even less, and to measure extraordinarily small changes
in wave-lengths, caused by different physical influences.

In addition to the continuous spectra emitted by
glowing solids or liquids, and to the line spectra emitted
by gases, and to the absorption spectra with dark lines,

LIGHT WAVES AND THE SPECTRUM

there are spectra of still another kind. These are the
absorption spectra which are produced by the passage
of white light through coloured glass or coloured fluids.
Here instead of fine dark lines there are broader dark
absorption bands, the spectrum being limited to the
individual bright parts. There are also the band spectra
proper, which, like the line spectra, are purely emission
spectra, given by the light from gases under particular
conditions ; these seem to consist of a series of bright

Fig. 12/ — Spectra produced by discharges of different character
through a glass tube containing nitrogen at a pressure of 1/20
that of the atmosphere. Above, a band spectrum ; below,
a line spectrum.

bands which follow each other with a certain regularity
(cf. Fig. 12). With stronger dispersion the bands are
* shown to consist of groups of bright lines.

Since the line spectra are most important in the
atomic theory, we shall examine them here more carefully.

The line spectra of the various elements differ very
much from each other with respect to their complexity.
While many metals give a great number of lines (iron,
for instance, gives more than five thousand), others give
only a few, at least in a simple spectroscope. With a
more powerful spectroscope the simplicity of structure

THE ATOM AND THE BOHR THEORY

is lost, since weaker lines appear and other lines which
had seemed single are now seen to be double or triple.
Moreover, the number of lines is increased by extending
the investigation to the ultra-violet and infra-red regions
of the spectrum. The sodium spectrum, at first, seemed
to consist of one single yellow line, but later this was
shown to be a double line, and still later several pairs of
weaker double lines were discovered. The kind and
number of lines obtained depends not only upon the
efficiency of the spectroscope, but also upon the physical
conditions under which the spectrum is obtained.

The eager attempts of the physicists to find laws
governing the distribution of the lines have been suc¬
cessful in some spectra. For instance, the line spectra
of lithium, sodium, potassium and other metals can
be arranged into three rows, each consisting of double
lines. The difference between the frequencies of the two
“ components ” of the double lines was found to be exactly
the same for most of the lines in one of these spectra, and
for the spectra of different elements there was discovered
a simple relationship between this difference in frequency
and the atomic weight of the element in question. But
this regularity was but a scrap, so to speak ; scientists were
still very far from a law which could exactly account for
the distribution of lines in a single series, not to mention
the lines in an entire spectrum or in all the spectra.

The first important step in this direction was made
about 1885 by the Swiss physicist, Balmer, in his investi¬
gations with the hydrogen spectrum, the simplest of all
the spectra. In the visible part there are just three
lines, one red, one green-blue and one violet, corre¬
sponding to the Fraunhofer lines C, F and h. These
hydrogen lines are now generally known by the letters

LIGHT WAVES AND THE SPECTRUM

Ha, H/i and H7. In the ultra-violet region there are
many lines also.

Balmer discovered that wave-lengths of the red and
of the green hydrogen line are to each other exactly as
two integers, namely, as 27 to 20, and that the wave¬
lengths of the green and violet lines are to each other as
28 to 25. Continued reflection on this correspondence
led him to enunciate a rule which can be expressed by
a simple formula. When frequency is substituted for
wave-length Balmer’s formula is written as

where v is the frequency of a hydrogen line, K a constant
equal to 3-29 xio15 and n an integer. If n takes on
different values, v becomes the frequency for the different
hydrogen lines. If n — 1 v is negative, for n= 2 v is
zero. These values of n therefore have no meaning with
regard to v. But if n— 3, then v gives the frequency for
the red hydrogen line Ha ; n= 4 gives the frequency of
the green line H^s and n= 5 that of the violet line H7.
Gradually more than thirty hydrogen lines have been
found, agreeing accurately with the formula for different
values of n. Some of these lines were not found in
experiment, but were discovered in the spectrum of
certain stars; the exact agreement of these lines with
Balmer’s formula was strong evidence for the belief
that they are due to hydrogen. The formula thus
proved itself valuable in revealing the secrets of the
heavens.

As n increases 1/w2 approaches zero, and can be made
as close to zero as desired by letting n increase indefinitely.
In mathematical terminology, as n— 00, i/w2=o and v —

THE ATOM AND THE BOHR THEORY

K/4 = 823 X io12, corresponding to a wave-length of 365 fjbfju.
Physically this means that the line spectrum of hydrogen
in the ultra-violet is limited by a line corresponding to
that frequency. Near this limit the hydrogen lines
corresponding to Balmer’s formula are tightly packed
together. For n =20 v differs but little from K/4, and the
distance between two successive lines corresponding to

7000 6000 6000 4000

1 1_ 1_ 1 ) J l! 1_ 1 J I I 1_ 1_ 1_ 1 I 1 i

1 1 t t V t 1 V

1 1

C F

I 1 1

He* %

Fig. 13. — Lines in the hydrogen spectrum corresponding

to the Balmer series.

an increase of I in n becomes more and more insignificant.
Fig. 13, where the numbers indicate the wave-lengths
in the Angstrom unit (o’l shows the crowding of
the hydrogen lines towards a definite boundary. The
following table, where K has the accurate value of
3*290364 x io15, shows how exactly the values calculated
from the formula agree with experiment.

Table of some of the Lines of the Balmer Series

I_i )~o (calculated).
V4 n V

v (found).

X (found).

n= 3

11 = 4
n= 5
n — 6
n= 7

K (l~ £ ) = 456,995 bills.

KU- iV) = 616,943 ,,

K( J— ^ ) =690,976 „
tfr) = 73i,*92 i>
K(l— tV) = 755>44° „

456,996 bills.
616,943 „
690,976 „
73L193 ,,
755,441 ,,

656*460 fx/A Ha
486 268 „
434*168 „ Hy
410*288 „ HS
397*119 „ He

n = 20

K(l-4k) = 8l4,365 »

814,361 „

368,307 „

LIGHT WAVES AND THE SPECTRUM

From arguments in connection with the work of the
Swedish scientist, Rydberg, in the spectra of other ele¬
ments, Ritz, a fellow countryman of Balmer’s, has made
it seem probable that the hydrogen spectrum contains
other lines besides those corresponding to Balmer’s
formula. He assumed that the hydrogen spectrum, like
other spectra, contains several series of lines and that
Balmer’s formula corresponds to only one series. Ritz
then enunciated a more comprehensive formula, the
Balmer-Ritz formula :

where K has the same value as before, and both n’ and
n" are integers which can pass through a series of different
values. For n" = 2, the Balmer series is given; to
n" = 1, and n' = 2, 3 . . . 00 there corresponds a second
series which lies entirely in the ultra-violet region, and
to n" =3, n' = 4, 5 . . . 00 a series lying entirely in the
infra-red. Lines have actually been found belonging to
these series.

Formulae, similar to the Ritz one, have been set up
for the line spectra of other elements, and represent
pretty accurately the distribution of the lines. The
frequencies are each represented by the difference
between two terms, each of which contains an integer,
which can pass through a series of values. But while
the hydrogen formula, except for the n’ s, depends only
upon one constant quantity K and its terms have the
simple form K In2, the formula is more complicated with
the other elements. The term can often be written,
with a high degree of exactness, as K/(n + a)2, where K
is, with considerable accuracy, the same constant as in

60 THE ATOM AND THE BOHR THEORY

the hydrogen formula. For a given element a can assume
several different values ; therefore the number of series
is greater and the spectrum is even more complicated
than that of hydrogen.

All these formulae are, however, purely empirical,
derived from the values of wave-lengths and frequencies
found in spectrum measurements. They represent
certain more or less simple bookkeeping rules, by which
we can register both old and new lines, enter them in
rows, arrange them according to a definite system.
But from the beginning there could be no doubt that
these rules had a deeper physical meaning which it was
not yet possible to know. There was no visible corre¬
spondence between the spectral line formulae and the
other physical characteristics of the elements which
emitted the spectra; not even in their form did the
formulae show any resemblance to formulae obtained in
other physical branches.

CHAPTER III

IONS AND ELECTRONS

✓

Early Theories and Laws of Electricity.

The fundamental phenomena of electricity, which
were first made the subject of careful study about two
centuries ago, are that certain substances can be
electrified by friction so that somehow they can attract
light bodies, and that the charges of electricity may
be either “ positive ” or “ negative.” Bodies with like
charges repel each other, while those with unlike charges
attract each other, and either partially or entirely
neutralize each other when they are brought close
together. Moreover, it had long ago been discovered
that in some substances electricity can move freely
from place to place, while in others there is resistance
to the movement. The former bodies are now called
conductors and include metals, while the latter are
called insulators, glass, porcelain and air being members
of this class.

In order to explain the phenomena some imagined
that there were two kinds of “ electric substances ” or
“ fluids ” ; and since no change in weight could be
discovered in a body when it was electrified, it was, in
general, assumed that the electric fluids were weight¬
less. In the normal, neutral body it was believed that

62 THE ATOM AND THE BOHR THEORY

these fluids were mixed in equal quantities, thereby
neutralizing each other; on this account they were
supposed to be of opposite characteristics, so one was
called positive and the other negative. According to a
second theory, there was assumed to be just one kind
of electricity, which was present in a normal amount
in neutral bodies ; positive electricity was caused by a
superfluity of the fluid ; negative, by a deficit. In both
theories it was possible to talk of the amount of positive
or negative electricity which a body contained or with
which it was “ charged,” because the supporters of the
one-fluid idea understood by the terms positive and
negative a superfluity and a deficit, respectively, of the
one fluid. In both theories it was possible to talk about
the direction of the electric current in a conductor,
since the supporters of the two-fluid theory understood
by “ direction ” that in which the electric forces sent
the positive electricity, or the opposite to that in which
the negative would be sent. It could not be decided
whether positive electricity went in the one direction
or the negative in the other, or whether each simultane¬
ously moved in its own direction. Both theories were
quite arbitrary in designating the electric charge in
glass, which was rubbed with woollen cloth, as positive.
On the whole, neither theory seemed to have any
essential advantage over the other ; the difference
between them seemed to lie more in phraseology than
in actual fact.

That the positive and negative states of electricity
could not be taken as “ symmetric ” seemed, however,
to follow from the so-called discharge phenomena, in
which electricity, with the emission of light, streams
out into the air from strongly charged (positive or

IONS AND ELECTRONS

negative) bodies, or passes through the air between
positive and negative bodies in sparks, electric arcs or in
some other way. In a discharge in air between a metal 1
point and a metal plate, for instance, a bush-shaped glow
is seen to extend from the point when the charge there
is positive, while only a little star appears when the
charge is negative.

Naturally, we cannot discuss here the many electric
phenomena and laws, and must be satisfied with a brief
description of those which are of importance in the
atomic theory. '

In this latter category belongs Coulomb's Law ,
formulated about 1785. According to this law, the
repulsions or attractions between two electrically
charged bodies are directly as the product of the charges
and inversely as the square of the distance between
them (as in the case of the gravitational attraction
between two neutral bodies, according to Newton’s
Law). The unit in measuring electric charges can be
taken as that amount which will repel an equal amount
of electricity of the same kind at unit distance with
unit force. If we use the scientific or “ absolute ”
system, in which the unit of length is one centimetre,
that of time one second and that of mass one gram,
then the unit of force is one dyne, which is a little greater
than the earth’s attraction on a milligram weight. Let
us suppose that two small bodies with equal charges
of positive (or negative) electricity are at a distance of
one centimetre from each other. If they repel each
other with a charge of one dyne, then the amount of
electricity with which each is charged is called the
absolute electrostatic unit of electricity. If one body has
a charge three times as great and the other has a charge

64 THE ATOM AND THE BOHR THEORY

four times as great, the repulsion is 3x4 = 12 times
greater. If the distance between the bodies is increased
from one to five, the repulsion is twenty-five times as
small, since 52=25. If the charge of one body is sub¬
stituted by a negative one of same magnitude the re¬
pulsion becomes an attraction of the same magnitude.

In the early part of the nineteenth century methods
were found for producing a steady electric current in
metal wires. In 1820, the Danish physicist, H. C.
0rsted, discovered that an electric current influences
a magnet in a characteristic way, and that, conversely,
the current is affected by the forces emanating from
the magnet, by a magnetic field in other words. The
French scientist, Ampere, soon afterwards formulated
exact laws for the electromagnetic forces between magnets
and currents. In 1831, the English physicist, Faraday,
discovered that an electric current is induced in a wire
when currents or magnets in its neighbourhood are
moved or change strength. Faraday’s views on electric
and magnetic fields of force around currents and magnets
were further of fundamental importance to the electro¬
magnetic-wave theory as developed by Maxwell. The
branch of physics dealing with all these phenomena is
now generally known as electrodynamics.

Electrolysis.

Faraday also studied the chemical effects which an
electric current produces upon being conducted between
two metal plates, called electrodes, which are immersed
in a solution of salts or acids. The current separates
the salt or acid into two parts which are carried by the
electric forces in two opposite directions. This separa¬
tion is called electrolysis. If the liquid is dilute hydro-

IONS AND ELECTRONS

chloric acid (HC1), the hydrogen goes with the current
to the negative electrode, the cathode , and takes the
positive electricity with it, while the chlorine goes
against the current and takes the negative electricity
to the positive electrode, the anode. We must then
assume with the Swedish scientist, Arrhenius, that,
under the influence of the water, the molecules of hydro¬
gen chloride always are separated into positive hydrogen
atoms and negative chlorine atoms, and that the electric

Fig. 14. — -Picture of electrolysis of hydrogen chloride.

A , anode ; K, cathode ; H , hydrogen atoms ; Cl, chlorine atoms.

forces from the anode and the cathode carry these
atoms respectively with and against the current. The
electrically charged wandering atoms are called ions,
i.e. wanderers. The positive electricity taken by the
hydrogen atoms to the cathode goes into the metal
conductor, while the anode must receive from the metal
conductor an equal amount of positive electricity to
be given to the chlorine atoms to neutralize them.
The negative charge of a chlorine atom must then be
as large as the positive charge of a hydrogen atom.

66 THE ATOM AND THE BOHR THEORY

These assumptions imply that equal numbers of the two
kinds of atoms are present in the whole quantity of
atoms transferred in any period of time.

Faraday found that the quantity of hydrogen which
in the above experiment is transferred to the cathode in
a given time is proportional to the quantity of electricity
transferred in the same time. A gram of hydrogen
always takes the same amount of electricity with it. By
experiment this amount of electricity can be deter¬
mined, and, since the weight in grams of the hydrogen
atom is known, it is possible to calculate the amount
of one atom. In electrostatic units it is 477 xio-10,
i.e.y 477 billionth1 parts. A chlorine atom then carries
with it 477 xio-10 electrostatic units of negative
electricity. Since its atomic weight is 35*5, then 35-5
grams of chlorine will take as much electricity as 1 gram
of hydrogen. The ratio e/m between the charge e and
the mass m is then 35*5 times as great for hydrogen as
for chlorine.

We have temporarily restricted ourselves to the
electrolysis of hydrogen chloride. Let us now assume
that we have chloride of zinc (ZnCl2), which, by elec¬
trolysis, is separated into chlorine and zinc. Each
atom of chlorine will, as before, carry 477 X io~10 units
of negative electricity to the anode ; but since zinc is
divalent (cf. p. 17) and one atom of zinc is joined to two
of chlorine, therefore one atom of zinc must carry a
charge of 2 X 477 X io~10 units of positive electricity to
the cathode. An atom or a group of atoms, with valence
of three, in electrolysis carries 3x477 Xio-10 units, etc.

We see then, that the quantity of electricity which

1 Billion used here to mean one million million, and trillion to mean
one million billion.

IONS AND ELECTRONS

accompanies the atoms in electrolysis is always
477Xio~10 electrostatic units or an integral multiple
thereof. This suggests the thought that electricity is
atomic and that the quantity 477 Xio-10 units is the
smallest amount of electricity which can exist in¬
dependently, i.e., the elementary quantum of electricity or
the “ atom of electricity.” The atom of a monovalent
element, when charged or ionized, should have one atom
of electricity; a divalent, two, etc. On the two-fluid
theory it was most reasonable to assume that there were

Fig. 15. — Provisional representation (according to the
two-fluid theory) of

A, a hydrogen ion ; B, a chlorine ion ; and C, a molecule of

hydrogen chloride.

two kinds of atoms of electricity representing, respec¬
tively, positive and negative electricity. In Fig. 15 there
is given, in accordance with the two-fluid theory, a rough
picture of a chlorine ion and a hydrogen ion and their
union into a molecule.

The atoms of electricity seemed to differ essentially
from the usual atoms of the elements in their apparent
inability to live independently ; they seemed to exist
only in connection with the atoms of the elements.
They would seem much more real if they could exist
independently. That such existence really is possible,

<,,0? <■/, /i>_/0^ 3*? X/t>'5 i. .

7 ‘ 68 THE ATOM AND THE BOHR THEORY

has been discovered by the study of the motion of
electricity in gases.

Vacuum Tube Phenomena.

It has previously been said that air is an insulator
for electricity, a statement which is, in general, true ;
however, as has also been said, electric sparks and arcs
can pass through air. Moreover, it has been discovered

that exhausted air is a very good conductor, so that a
strong current can pass between two metal electrodes
in a glass tube where the air is exhausted, if the electrodes
are connected to an outer conductor by metal wires
fused into the glass. In these vacuum tubes there are
produced remarkable light effects, at first inexplicable.
When the air is very much exhausted, to a hundred
thousandth of the atmospheric pressure or less, strong
electric forces (large difference of potential between the
electrodes) are needed to produce an electric discharge.

Yig. 1 6. _ Vacuum tube with cathode rays and a shadow-

producing cross.

P and N, conducting wires for the electric current ; a, cathode ;
b, anode and shadow-producer ; c, d, the shadow.

IONS AND ELECTRONS

Such a discharge assumes an entirely new character ; in
the interior of the glass tube there is hardly any light
to be seen, but the glass wall opposite the negative
electrode (the cathode) glows with a greenish tint
(fluorescence). If a small metal plate is put in the tube
between the cathode and the glass wall, a shadow is cast
on the wall, just as if light were produced by rays,
emitted from the cathode at right angles to its surface
(cf. Fig. 16). The English physicist, Crookes, was one
of the first to study these cathode rays. He assumed
that they are not ether waves like the light rays, but

Fig. 17. — Vacuum tube, where a bundle of cathode rays are
deviated by electric forces.

A, anode ; K, cathode.

that they consist of particles which are hurled from the
cathode with great velocity in straight lines ; they light
the wall by their collisions with it. There was soon no
doubt as to the correctness of Crookes’ theory. The
cathode rays are evidently particles of negative elec¬
tricity, which by repulsions are driven from the cathode
(the negative electrode). A metal plate bombarded
by the rays becomes charged negatively. Let us
suppose that we have a small bundle of cathode rays,
obtained by passing the rays from the cathode K
(cf. Fig. 17) through two narrow openings Sx and 5.
It can then be shown that the bundle of rays is deviated

70 THE ATOM AND THE BOHR THEORY

not only by electric forces, but also by magnetic action
from a magnet which is held near the glass. In the
figure there is shown a deviation of the kind mentioned,
caused by making the plates at B and C respectively
positive and negative ; since B attracts the negative
particles and C repels them, the light spot produced by
the bundle of rays is moved from M to Mv The magnetic
deviation is in agreement with 0rsted’s rules for the
reciprocal actions between currents and magnets, if we
consider the bundle of rays produced by moving electric
particles as an electric current. (Since the electric
particles travelling in the direction of the rays are
negative, and since it is customary by the expression
“ direction of current ” to understand the direction
opposite to that in which the negative electricity moves,
then, in the case of the cathode rays just mentioned, the
direction of the current must be opposite to that of the
rays.)

From measurements of the magnetic and electric
deviations it is possible to find not only the velocity of
the particles, but also the ratio e/m between the charge e
of the particle and its mass m. The velocity varies with
the potential at the cathode, and may be very great,
50,000 km. per second, for instance (about one-sixth the
speed of light), or more. It has been found that ejm
always has the same value, regardless of the metal of
the cathode and of the gas in the tube ; this means that
the particles are not atoms of the elements, but some¬
thing quite new. It has also been found that e/ m is about
two thousand times as large as the ratio between the
charge and the mass of the hydrogen atom in electro¬
lysis. If we now assume that e is just the elementary
quantum of electricity 477X10*-10, which in magnitude

IONS AND ELECTRONS

amounts to the charge of the hydrogen atom in elec¬
trolysis (but is negative), then m must have about
1/2000 the mass of the hydrogen atom. This assump¬
tion as to the size of e has been justified by experiments
of more direct nature. The experiments with charge
and mass of electrons which have in particular been
carried out by the English physicist, J. J. Thomson,
give reason then to suppose these quite new and unknown
particles to be free atoms of negative electricity ; they
have been given the name of electrons. Gradually more
information about them has been acquired. Thus it
has been possible in various ways to determine directly
the charge on the electron, independently of its mass.
Special mention must be made of the brilliant investiga¬
tions of the American, Millikan, on the motion of very
small electrified oil-drops through air under the influence
of an electric force. To Millikan is due the above-
mentioned value of e, which is accurate to one part in
five hundred. Further, the mass of the electron has been
more exactly calculated as about 1/1835 that the
hydrogen atom. Their magnitude has also been learned ;
the radius of the electron is estimated as 1-5 xio-13
cm. or 1-5x10 an order of magnitude one ten-
thousandth that of the molecule or atom.

After the atom of negative electricity had been
isolated, in the form of cathode rays, the next suggestion
was that corresponding positive electric particles might
be discharged from the anode in a vacuum tube. By
special methods success has been attained in showing
and studying rays of positive particles. In order to
separate them from the negative cathode ray particles
the German scientist, Goldstein, let the positive particles
pass through canals in the cathode ; they are therefore

72 THE ATOM AND THE BOHR THEORY

called canal rays. The velocity of the particles is much
less than that of the cathode rays, and the ratio e/m
between charge and mass is much smaller and varies
according to the gas in the tube. In experiments where
the tube contains hydrogen, rays are always found for
which e/m, as in electrolysis, is about 1/2000 of the ratio
in the cathode rays. Therefore there can be scarcely
any doubt that these canal rays are made up of charged
hydrogen atoms or hydrogen ions. The values found
with other gases indicate that the particles are atoms
(or molecules sometimes) of the elements in question,
with charges one or more times the elementary quantum
of electricity (477 X nr10 electrostatic units). Research
in this field has also been due in particular to J. J. Thom¬
son. From his results, as well as from those obtained
by other methods, it follows that positive electricity,
unlike negative, cannot appear of its own accord, but is
inextricably connected to the atoms of the elements.

The Nature of Electricity.

The earlier conceptions of a one or two-fluid explana¬
tion of the phenomena of electricity appear now in a
new light. We are led to think of a neutral atom as
consisting of one mass charged with positive electricity
together with as many electrons negatively charged as
are sufficient to neutralize the positive. If the atom
loses one, two or three electrons, it becomes positive with
a charge of one, two or three elementary quanta of
electricity, or for the sake of simplicity and brevity we
say that the atom has one, two or three “ charges.” If,
on the other hand, it takes up one, two or three extra
electrons it has one, two or three negative charges.
Fig. 18 can give help in understanding these ideas, but

IONS AND ELECTRONS

it must not be thought that the electrons are arranged
in the way indicated. The substances, which appear as
electropositive in electrolysis — i.e. hydrogen and metals
—should then be such that their atoms easily lose one
or more electrons, while the electronegative elements
should, on the other hand, easily take up extra electrons.
Elements should be monovalent or divalent according
as their atoms are apt to lose or to take up one or two

FIG. 1 8. — Provisional representation (according to the electron

theory) of

A, a neutral atom ; B, the same atom with two positive charges
(a divalent positive ion) and C, the same atom with two negative
charges (a divalent negative ion).

electrons. From investigations with the vacuum tube
it appears, however, that the atoms of the same element
can in this respect behave in more ways than would be
expected from electrolysis or chemical valence.

When an electric current passes through a metal wire,
it must be assumed that the atoms of the metal remain
in place, while the electrical forces carry the electrons
in a direction opposite to that which usually is con¬
sidered as the direction of the current (cf. p. 70). The
motion of the electrons must not be supposed to proceed
without hindrance, but rather as the result of a com-

THE ATOM AND THE BOHR THEORY

plicated interplay, by no means completely understood,
whereby the electrons are freed from and caught by the
atoms and travel backwards and forwards, in such a
way that through every section of the metal wire a
surplus of electrons is steadily passing in the direction
opposite to the so-called direction of the current. The
number of surplus electrons which in every second passes
through a section of the thin metal wire in an ordinary
twenty-five candle incandescent light, at 220 volts,
amounts to about one trillion (io18), or 1000 million
(io9) in 0*000,000,001 of a second. If the metal con¬
ducting wire ends in the cathode of a vacuum tube, the
electrons carried through the wire pass freely into the
tube as cathode rays from the cathode. *

This motion of electricity agrees best with the one-
fluid theory, since the electrons, which here alone accom¬
plish the passage of the electricity, may be considered
as the fundamental parts of electricity. In this respect
the choice of the terms positive and negative is very
unfortunate, since a body with a negative charge actually
has a surplus of electrons. Moreover, the electrons
really have mass ; but since th§ mass of a single electron
is only 1/1835 that of the atom of the lightest element,
hydrogen, and since in an electrified body which can be
weighed by scale there is always but an infinitesimal
number of charged atoms, it is easy to understand that,
formerly, electricity seemed to be without weight.

In electrolysis, where the motion of electricity is
accomplished by positive and negative ions, we have a
closer connection with the two-fluid theory. In motions
of electricity through air the situation suggests both the
one-fluid and the two-fluid theories, since the passage of
electricity is sometimes carried on exclusively by the

IONS AND ELECTRONS

electrons, and sometimes partly by them and partly
by larger positive and negative ions, i.e., atoms or mole¬
cules with positive and negative charges.

The Electron Theory.

Proceeding on the assumption that the electric and
optical properties of the elements are determined by the
activity of the electric particles, the Dutch physicist
Lorentz and the English physicist Larmour succeeded in
formulating an extraordinarily comprehensive “ electron
theory,” by which the electrodynamic laws for the
variations in state of the ether were adapted to the
doctrine of ions and electrons. This Lorentz theory must
be recognized as one of the finest and most significant
results of nineteenth century physical research.

It was one of the most suggestive problems of this
theory to account for the emission of light waves from
the atom. From the previously described electro¬
magnetic theory of fight (cf. p. 42) it follows that an
electron oscillating in an atom will emit fight waves
in the ether, and that the frequency v of these waves will
naturally be equal to the number of oscillations of the
electron in a second. If this last quantity is designated
as a, then

V—QO

It may then be supposed that the electrons in the un¬
disturbed atom are in a state of rest, comparable with
that of a ball in the bottom of a bowl. When the atom
in some way is “ shaken,” one or more of the electrons
in the atom begins to oscillate with a definite frequency,
just as the ball might roll back and forth in the bowl if
the bowl was shaken. This means that the atom is
emitting fight waves, which, for each individual electron

76 THE ATOM AND THE BOHR THEORY

have a definite wave-length corresponding to the fre¬
quency of the oscillations, and that, in the spectrum of
the emitted light, the observed spectral lines correspond
to these wave-lengths.

Strong support for this view was afforded by Zeeman’s
discovery of the influence of a magnetic field upon spectral
lines. Zeeman, a Dutch physicist, discovered, about
twenty-five years ago, that when a glowing vacuum
tube is placed between the poles of a strong electro¬
magnet, the spectral lines in the emitted light are split
so that each line is divided into three components with
very little distance between them. It was one of the
great triumphs of the electron theory that Lorentz was
able to show that such an effect was to be expected if
it was assumed that the oscillations of fight were pro¬
duced by small oscillating electric particles within the
atom. From the experiments and from the known laws
concerning the reciprocal actions of a magnet and an
electric current (here the moving particle), the theory
enabled Lorentz to find not only the ratio e/m between
the electric charge of each of these particles and its mass,
but also the nature of the charge. He could conclude
from Zeeman’s experiment that the charge is negative
and that the ratio e/m is the same as that found for the
cathode rays. After this there could not well be doubt
that the electrons in the atoms were the origin of the
fight which gives the fines of the spectrum. It seemed,
however, quite unfeasible for the theory to explain the
details in a spectrum — to derive, for instance, Balmer’s
formula, or to show why hydrogen has these fines,
copper those, etc. These difficulties, combined with
the great number of fines in the different spectra,
seemed to mean that there were many electrons in

IONS AND ELECTRONS

an atom and that the structure of an atom was ex¬
ceedingly complicated.

Ionization by X-rays and Rays from Radium. Radio¬
activity.

As has been said, the electrons in a vacuum tube
cause its wall to emit a greenish light when they strike
it. Upon meeting the glass wall or a piece of metal
(the anticathode) placed in the tube the electrons cause
also the emission of the peculiar, penetrating rays
called Rontgen rays in honour of their discoverer, or
more commonly X-rays. They may be described as
ultra-violet rays with exceedingly small wave-lengths
(cf. p. 54). When, further, the electrons meet gas
molecules in the tube they break them to pieces, separat¬
ing them into positive and negative ions {ionization).
The positive ions are the ones which appear in the canal
rays. The ions set in motion by electrical forces can
break other gas molecules to pieces, thus assisting in
the ionization process. At the same time the gas
molecules and atoms are made to produce disturbances
in the ether, and thus to cause the light phenomena •
which arise in a tube which is not too strongly exhausted.

The free air can be ionized in various ways ; this
ionization can be detected because the air becomes
more or less conducting. In fact, electric forces will
drive the positive and negative ions through the air
in opposite directions, thus giving rise to an electric
current. If the ionization process is not steadily
continued, the air gradually loses its conductivity, since
the positive and negative ions recombine into neutral
atoms or molecules. Ionization can be produced by
flames, since the air rising from a flame contains ions.

78 THE ATOM AND THE BOHR THEORY

A strong ionization can also be brought about by X-rays
and by ultra-violet rays. In the higher strata of the
atmosphere the ultra-violet rays of the sun exercise
an ionizing influence. Most of all, however, the air is
ionized by rays from the so-called radioactive substances
which in very small quantities are distributed about
the world.

The characteristic radiation from these substances
was discovered in the last decade of the nineteenth
century by the French physicist, Becquerel, and after¬
wards studied by M. and Mme. Curie. From the radio¬
active uranium mineral, pitchblende, the latter separated
the many times more strongly radioactive element
radium. The proper nature of the rays was later
explained, particularly through the investigations of the
English physicists, Rutherford, Soddy and Ramsay.
These rays, which can produce heat effects, photographic
effects and ionization, are of three quite different
classes, and accordingly are known as a-rays, |3-rays,
and y-rays. The last named, like the X-rays, are
ultra-violet rays, but they have often even shorter wave¬
lengths and a much greater power of penetration than
the usual X-rays. The |3-rays are electrons which are
ejected with much greater velocity than the cathode
rays ; in some cases their velocity goes up to 99-8
per cent, that of light. The a-rays are positive atomic
ions, which move with a velocity varying according to
the emitting radioactive element from 1/20 to almost
1/10 that of light. It has further been proved that the
a-particles are atoms of the element helium, which
has the atomic weight 4, and that they possess two
positive charges, i.e., they must take up two electrons
to produce a neutral helium atom.

IONS AND ELECTRONS

There is no doubt that the process which takes place
in the emission of radiation from the radioactive
elements is a transformation of the element, an explosion
of the atoms accompanied by the emission either of
double-charged helium atoms or of electrons, and the
forming of the atoms of a new element. The energy of
the rays is an internal atomic energy, freed by these
transformations. The element uranium, with the
greatest of all known atomic weights (238), passes, by
several intermediate steps, into radium with atomic
weight 226 ; from radium there comes, after a series of
steps, lead, or, in any case, an element which, in all its
chemical properties, behaves like lead. We shall go no
further into this subject, merely remarking that the
transformations are quite independent of the chemical
combinations into which the radioactive elements
have entered, and of all external influences.

When a-particles from radium are sent against a
screen with a coating of especially prepared zinc sulphide,
on this screen, in the dark, there can be seen a character¬
istic light phenomenon, the so-called scintillation, which
consists of many flashes of light. Each individual
flash means that an a-particle, a helium atom, has hit
the screen. In this bombardment by atoms the in¬
dividual atom-projectiles are made visible in a manner
similar to that in which the individual raindrops which
fall on the surface of a body of water are made visible
by the wave rings which spread from the places where
the drops meet the water. This flash of light was the
first effect of the individual atom to be available for
investigation and observation. The incredibility of
anything so small as an atom producing a visible effect
is lessened when, instead of paying attention merely

THE ATOM AND THE BOHR THEORY

to the small size or mass of the atom, its kinetic energy
is considered ; this energy is proportional to the square
of the velocity, which is here of overwhelming magnitude.
For the most rapid a-particles the velocity is 2*26 xio9
cm. per second ; their kinetic energy is then about

Fig. 19. — Photograph of paths described by
a-particles (positive helium ions) emitted
from a radioactive substance.

1/30 of the kinetic energy of a weight of one milligram
of a substance at a velocity of one centimetre per second.
This energy may seem very small, but, at least, it is
not a magnitude of “ inconceivable minuteness,” and
it is sufficient under the conditions given above to produce
a visible light effect. We must here also consider the
extreme sensitiveness of the eye.

IONS AND ELECTRONS

More practical methods of revealing the effects of
the individual a-particles and of counting them are
founded on their very strong ionization power. By
strengthening the ionization power of a-particles,
Rutherford and Geiger were able to make the air in a

Fig. 20. — Photograph of the path of a /3-particle

(an electron) .

(Both 19 and 20 are photographs by C. T. R. Wilson.)

so-called ionization chamber so good a conductor that
an individual a-particle caused a deflection in an
electrical apparatus, an electrometre.

With a more direct method the English scientist
C. T. R. Wilson has shown the paths of the a-particles
by making use of the characteristic property of ions,
that in damp air they attract the neutral water mole-
6

THE ATOM AND THE BOHR THEORY

cules which then form drops of water with the ions as
nuclei. In air which is completely free of dust and
ions the water vapour is not condensed, even if the
temperature is decreased so as to give rise to super¬
saturation, but as soon as the air is ionized the vapour
condenses into a fog. When Wilson sent a-particles
through air, supersaturated with water vapour, the
vapour condensed into small drops on the ions produced
by the particles ; the streaks of fog thus obtained
could be photographed. Fig. 19 shows such a photo¬
graph of the paths of a number of atoms. When a
streak of fog ends abruptly it does not mean that the
a-particles have suddenly halted, but that their velocity
has decreased so that they can no longer break the
molecules of air to pieces, producing ions. The paths
of the /3-particles have been photographed in the same
way, although an electron of the ,8-particles has a mass
about 7000 times smaller than that of a helium atom ;
the electron has, however, a far greater velocity than
the helium atom. This velocity causes the ions to be
farther apart, so that each drop of water formed around
the individual ions can appear in the photograph by
itself (cf. Fig. 20).

CHAPTER IV
THE NUCLEAR ATOM

Introduction.

We are now brought face to face with the funda¬
mental question, hardly touched upon at all in the
previous part of this work, namely, that of the con¬
struction and mode of operation of the atomic mechanism
itself. In the first place we must ask : What is the
“ architecture ” of the atom, that is, what positions
do the positive and negative particles take up with
respect to each other, and how many are there of each
kind ? In the second place, of what sort are the pro¬
cesses which take place in an atom, and how can we
make them interpret the physical and chemical pro¬
perties of the elements ? In this chapter we shall keep
essentially to the first question, and consider especially
the great contribution which Rutherford made in 1911
to its answer in his discovery of the positive atomic
nucleus and in the development of what is known as the
Rutherford atomic model or nuclear atom.

Rutherford’s Atom Model.

Rutherford’s discovery was the result of an in¬
vestigation which, in its main outlines, was carried out
as follows : a dense stream of a-particles from a powerful

THE ATOM AND THE BOHR THEORY

radium preparation was sent into a highly exhausted
chamber through a little opening. On a zinc sulphide
screen, placed a little distance behind the opening, there
was then produced by this bombardment of atomic
projectiles, a small, sharply defined spot of light. The
opening was next covered by a thin metal plate, which
can be considered as a piece of chain mail formed of
densely-packed atoms. The a-particles, working their
way through the atoms, easily traversed this “ piece
of mail ” because of their great velocity. But now it
was seen that the spot of light broadened out a little
and was no longer sharply limited. From this fact one
could conclude that the a-particles in passing among the
many atoms in the metal plate suffered countless, very
small deflections, thus producing a slight spreading of
the rays. It could also be seen that some, though
comparatively few, of the a-particles broke utterly
away from the stream, and travelled farther in new
directions, some, indeed, glancing back from the metal
plate in the direction in which they had come (cf. Fig. 21).
The situation was approximately as if one had dis¬
charged a quantity of small shot through a wall of
butter, and nearly all the pellets had gone through the
wall in an almost unchanged direction, but that one or
two individual ones had in some apparently uncalled-
for fashion come travelling back from the interior of the
butter. One might naturally conclude from this cir¬
cumstance that here and there in the butter were located
some small, hard, heavy objects, for example, some
small pellets with which some of the projectiles by chance
had collided. Accordingly, it seemed as if there were
located in the metal sheet some small hard objects.
These could hardly be the electrons of the metal atoms,

THE NUCLEAR ATOM

because a-particles, as has been stated before, are helium
atoms with a mass over seven thousand times that of a
single electron ; and if such an atom collided with an
electron, it would easily push the electron aside without
itself being deviated materially in its path. Hardly any
other possibility remained than to assume that what

Fig. 21. — Tracks of a-particles in the interior of matter. While

1 and 3 undergo small deflections by collisions with electrons,

2 is sharply deflected by a positive nucleus.

the a-particles had collided with was the positive part
of the atom, whose mass is of the same order of magni¬
tude as the mass of the helium atom (cf. Fig. 21).
A mathematical investigation showed that the large
deflections were produced because the a-particles in
question had passed, on their way, through a tre¬
mendously strong electric field of the kind which will
exist about an electric charge concentrated into a very
small space and acting on other charges according to
Coulomb’s Law. When, in the foregoing, the word

THE ATOM AND THE BOHR THEORY

" collision ” is used, it must not be taken to mean simply
a collision of elastic spheres; rather the two particles
(the a-particle and the positive particle of the metal
atom) come so near to each other in the flight of the
former that the very great electrical forces brought into
play cause a significant deflection of the a-particles from
their original course.

Rutherford was thus led to the hypothesis that
nearly all of the mass of the atom is concentrated into
a positively charged nucleus, which, like the electrons, is
very small in comparison with the size of the whole
atom ; while the rest of the mass is apportioned among
a number of negative electrons which must be assumed
to rotate about the nucleus under the attraction of the
latter, just as the planets rotate about the sun. Under
this hypothesis the outer limits of the atom must be
regarded as given by the outermost electron orbits.
The assumption of an atom of this structure makes it at
once intelligible why, in general, the a-particles can
travel through the atom without being deflected materi¬
ally by the nuclear repulsion, and why the very great
deflections occur as seldom as is indicated by experiment.
This latter circumstance has, on the other hand, no
explanation in the atomic model previously suggested
by Lord Kelvin and amplified by J. J. Thomson, in which
the positive electricity was assumed to be distributed
over the whole volume of the atom, while the electrons
were supposed to move in rings at varying distances
from the centre of the atom.

The same characteristic phenomenon made evident
in the passage of a-particles through substances by the
investigations of Rutherford appears in a more direct
way in Wilson’s researches discussed on p. 81. His

THE NUCLEAR ATOM

photographs of the paths of a-particles through air super¬
saturated with water vapour (see Fig. 22) show pro¬
nounced kinks in the paths of individual particles.
Thus in the figure referred to, there are shown the paths
of two a-particles. One of these is almost a straight
line (with a very slight curvature), while the other'
shows a very perceptible deflection as it approaches
the immediate neighbourhood of the nucleus of an atom,
and finally a very abrupt kink ; at the latter place it is
clear that the a-particle has penetrated very close to the
nucleus. If one examines the picture more closely ^

Fig. 22. — Photograph of the paths of two a-particles (positive
helium ions). One collides with an atomic nucleus.

there will be seen a very small fork at the place where
the kink is located. Here the path seems to have
divided into two branches, a shorter and a longer. This
leads one at once to suspect that a collision between two
bodies has taken place, and that after the collision each
body has travelled its own path, just as if, to return to
the analogy of the bombardment of the butter wall, one
had been able to drive two pellets out of the butter by
shooting in only one. Or, to take perhaps a more familiar
example, when a moving billiard ball collides at random
with a stationary one, after the collision they both
move off in different directions. So, when the a-particle

88 THE ATOM AND THE BOHR THEORY

hits at random the atomic nucleus, both particle and
nucleus move off in different directions ; though in this
case, since the nucleus has the much greater mass of the
two, it moves more slowly, after the collision, than the
a-particle, and has, therefore, a much shorter range
in the air than the lighter, swifter a-particle. Had
the gas in which the collisions took place been
hydrogen, for example, the recoil paths of the hydro¬
gen nuclei would have been longer than those of the
a-particles, because the mass of the hydrogen nucleus
is but one quarter the mass of the a-particle (helium
atom).

The collision experiments on which Rutherford’s
theory is founded are of so direct and decisive a character
that one can hardly call it a theory, but rather a fact,
founded on observation, showing conclusively that the
atom is built after the fashion indicated. Continued
researches have amassed a quantity of important facts
about atoms. Thus, Rutherford was able to show that
the radius of the nucleus is of the order of magnitude
io-12 to io-13 cm. This means really that it is only when
an a-particle approaches so near the centre of an atom
that forces come into play which no longer follow
Coulomb’s Law for the repulsion between two point
charges of the same sign (in contrast to the case in the
ordinary deflections of a-particles) . It should be re¬
marked, however, that in the case of the hydrogen
nucleus theoretical considerations give foundation for
the assumption that its radius is really many times
smaller than the radius of the electron, which is some
2000 times lighter ; experiments by which this assump¬
tion can be tested are not at hand at present.

THE NUCLEAR ATOM

The Nuclear Charge ; Atomic Number ; Atomic Weight.

It is not necessary to have recourse to a new research
to determine the masses of the nuclei of various atoms,
because the mass of the nucleus is for all practical pur¬
poses the mass of the atom. Accordingly, if the mass of
the hydrogen nucleus is taken as unity, the atomic mass
is equal to the atomic weight as previously defined.
The individual electrons which accompany the nucleus
are so light that their mass has relatively little influence
(within the limits of experimental accuracy) on the total
mass of the atom.

On the other hand, a problem of the greatest import¬
ance which immediately suggests itself is to determine
the magnitude of the positive charge of the nucleus.
This naturally must be an integral multiple of the
fundamental quantum of negative electricity, namely,
477 X io~10 electrostatic units, or if we prefer to call this
simply the “ unit ” charge, then the nuclear charge must
be an integer. Otherwise a neutral atom could not be
formed of a nucleus and electrons, for in a neutral atom
the number of negative electrons which move about
the nucleus must be equal to the number of positive
charges in the nucleus. The determination of this num¬
ber is, accordingly, equivalent to the settling of the
important question, how many electrons surround the
nucleus in the normal neutral state of the atom of the
element in question.

The answer to the question is easiest in the case of the
helium atom. For when this is expelled as an a-particle,
it carries, as Rutherford was able to show, a positive
charge of two units — in other words, two electrons are
necessary to change the positive ion into a neutral atom.

THE ATOM AND THE BOHR THEORY

At the same time there is every reason to suppose that
the a-particle is simply a helium nucleus deprived of its
electrons ; it follows, therefore, that the electron system
of the neutral helium atom consists of two electrons.
Since the atomic weight of helium is four, the number of
electrons is consequently one-half the atomic weight.
Rutherford’s investigation of the deflections of a-particles
in passing through various media had already led him
to believe that for many other elements, to a consider¬
able approximation, the nuclear charge and hence the

ABC

Fig. 23. — Schematic representation of the nuclear atom.

A, a neutral hydrogen atom ; B, a positive, and C, a negative hydrogen

ion ; K, atomic nuclei ; E, electrons.

' \

number of electrons was equal to half the atomic weight.
Hydrogen, of course, must form an exception, since its
atomic weight is unity. The positive charge on the hydro¬
gen nucleus is one elementary quantum, and in the neutral
state of the atom, only one electron rotates about it. Fig. 23
gives a representation of the structure of the hydrogen
atom, and the structures of the two types of hydrogen
ions formed respectively by the loss and gain of an elec¬
tron. In the picture, the position of the electron is, of
course, arbitrary, and for the sake of simplicity its path
is supposed to be circular.

THE NUCLEAR ATOM

As has just been indicated, Rutherford’s rule for the
number of electrons is only an approximation. A Dutch
physicist, van den Broek, conceived in the meantime
the idea that the number of electrons in the atom of an
element is equal to its order number in the periodic
table (its " atomic number,” as it is now called).
Especially through a systematic investigation of the
X-ray spectra characteristic of the different elements
this has proved to be the correct rule. In fact, using
Bragg’s reflection method of X-rays from crystal sur¬
faces (cf. p. 54), the Englishman, Moseley, made in 1914
the far-reaching discovery that these spectra possess an
exceptionally simple structure, which made it possible
in a simple way to attach an order number to each
element (given on p. 23). On the basis of Bohr’s
theory, established a year before, it could be directly
proved that this order number must be identical
with the number of positive elementary charges on
the nucleus.

The number which formerly indicated simply the
position of an element in the periodic system has thus
obtained a profound physical significance, and in com¬
parison the atomic weight has come to have but a secon¬
dary meaning. The inversion of argon and potassium
in the periodic system (mentioned on p. 21), which seemed
to be an exception to the regularity displayed by the
system as a whole, obtains an easy explanation on the
van den Broek rule ; for to explain the inversion we need
only assume that potassium has one electron more than
argon, though its atomic weight is less than that of argon.
We see at once that the atomic weight and number of
electrons (or what is the same thing — the nuclear
charge) are not directly correlated to each other.

92 THE ATOM AND THE BOHR THEORY

And since the periodic system based on the atomic
number represents the correct arrangement of the
elements according to their respective properties (es¬
pecially their chemical properties), we are led natur¬
ally to the conclusion that it is the atomic number
and not the atomic weight that determines chemical
characteristics.

The conception of the relatively great importance of
the atomic number as compared with the atomic weight
has in recent years received overwhelming support from
the researches of Soddy, Fajans, Russell, Hevesy and
others who have discovered the existence of so-called
isotope elements (from the Greek isos= same, and topos =
place), substances with different nuclear masses (atomic
weights) and different radioactive properties (if there
are any), but with the same nuclear charge, the same
number of electrons and, consequently, occupying the
same place in the periodic system. Two such isotopes
are practically equivalent in all their chemical properties
as well as in most of their physical characteristics. One
of the oldest examples of isotopes is provided by ordinary
lead with the atomic weight 207*2 and the substance
found in pitchblende with the atomic weight 206, but
identical, chemically, with ordinary lead. This latter
form of lead has already been referred to on p. 79 as the
end product of radioactive disintegrations, and hence it
is sometimes called radium lead.

By his investigations of canal rays the English
physicist Aston has just recently shown that many
substances which have always been assumed to be simple
elements, are in reality mixtures of isotopes. The
atomic weight of chlorine determined in the usual way
is 35*5, but in the discharge tube two kinds of chlorine

THE NUCLEAR ATOM

atoms appear, having atomic weights 35 and 37 respec¬
tively ; and it must be assumed that these two kinds of
chlorine are present in all the compounds of chlorine
known on the earth in the ratio of, roughly, three to one.
To separate such mixtures into their constituent parts is
extremely difficult, precisely because the constituents
have identical properties apart from a small difference
in density, which stands in direct connection with the
atomic weight. Such a separation was first carried out
successfully by the Danish chemist, Br0nsted, in collabo¬
ration with the Hungarian chemist, Hevesy (1921).
These two scientists were able to separate a large quantity
of mercury of density 13*5955 into two portions of slightly
different densities. All the different isotopes of which
mercury is a mixture were, indeed, not wholly separated ;
they were represented in the two portions in different
proportions. Thus, in one of the first attempts, the
density of the one part was 13-5986 and of the other
13-5920 (at o° C).

It is a perfectly reasonable supposition that it is
the electron system which determines the external
properties of the atom, that is, those properties which
depend on the interplay of two or more atoms. For
the electron, rotating about the nucleus at a considerable
distance, separates, so to speak, the nucleus from the
surrounding space, and must therefore be assumed to
be the organ which connects the atom with the rest of
the universe. One might also expect the structure of
the electron system to depend wholly on the nuclear
charge, i.e. on the atomic number and not on the mass of
the nucleus, since it is the nuclear electrical attraction
which holds the electrons in their orbits and not the
relatively insignificant gravitational attraction.

94 THE ATOM AND THE BOHR THEORY

It thus becomes intelligible that the properties of
the elements can be divided into two sharply defined
classes, namely : (i) properties of the nucleus, and

(2) properties of the electron system in the atom.
The credit for first recognizing the sharp distinction
between these two classes, a distinction fundamental
for a detailed study of the atom, is due to Niels
Bohr.

The properties of the nucleus determine— (a) the
radioactive processes, or explosions of the nucleus, and
related processes ; ( b ) collisions, where two nuclei

approach extremely near to each other ; and (c) weight
which, as mentioned above, stands in direct connection
with atomic weight. The properties of the electron
system are, on the other hand, the determining factors
in all other physical and chemical activities, and, as
has been stated, are functions, we may say, of the
atomic number of the given element. The Bohr theory
may be said to concern itself with the chemical and
physical properties of the atom with the exception of
those which have to do with the nucleus. We shall
consequently devote our attention in the next chapters
to the electron system. But before turning to this
we shall dwell a little further upon the atomic
nucleus.

The Structure of the Nucleus.

That the nucleus is not an elementary indivisible
particle but a system of particles, is clearly shown by
the radioactive processes in which a-particles and f3-
particles (electrons) are shot out of the nuclei of radio¬
active elements. Bohr was the first to see clearly that

THE NUCLEAR ATOM

not only the a-particles emitted in such cases come
from the nucleus, but that the |3-particles also have
their source there. There is now no doubt that, in
addition to the outer electrons of the atom, which are
the determining factor in the atomic number, there
must also be, in the radioactive substances at any rate,
special nuclear electrons which lead a more hidden
existence in the interior of the nucleus. One can easily
understand that isotopes may result as products of
radioactive disintegration. For example, let us suppose
that a nucleus emits first an os-particle (i.e. a helium
nucleus with two positive charges), and thereafter sends
out two electrons, each with its negative charge, in
two new disintegrations. The nuclear charge in the
resultant atom will then obviously be the same as
before, because the loss of the two electrons exactly
neutralizes that of the a-particle. - But the atomic
weight will be diminished by four units (i.e. the weight
of the helium nucleus, remembering also that the
electrons have but very negligible masses). Among
the radioactive substances are recognized many
examples of isotope elements, with atomic weights
differing precisely by four. The radioactive element
uranium is the element with the greatest atomic weight
(238), and atomic number (92), and consequently with
the greatest nuclear charge. Almost all the other radio¬
active substances are those with high atomic numbers
in the periodic system. The cause of radioactivity
must be sought in the hypothesis that the nuclei of the
radioactive elements are very complicated systems with
small stability, and therefore break down rather easily
into less complicated and more stable systems with
the emission of some of their constituent particles ;

96 THE ATOM AND THE BOHR THEORY

the corpuscular rays thus produced possess a consider¬
able amount of kinetic energy.

Accordingly, by analogy, the nuclei of the non¬
radioactive elements may be assumed to be composed
of nuclear electrons and positive particles ; hydrogen
alone excepted. The simplest assumption is that the
hydrogen nucleus is the real quantum or atom of positive
electricity, just as the electron is the atom of negative
electricity. On this theory all substances are built up
of two kinds only of fundamental particles, namely,
hydrogen nuclei and electrons. That these particles
may themselves consist of constituent parts is, of
course, an open possibility, but such speculation is
beyond our experience up to the present. In every
nucleus there are more positive hydrogen nuclei than
there are negative electrons, so that the nucleus has a
residual positive charge of a magnitude equal to the
difference between the number of hydrogen nuclei and
nuclear electrons.

If we now pass from hydrogen which has the atomic
weight, atomic number and nuclear charge of unity, we
next encounter helium with the atomic weight 4, atomic
number and nuclear charge 2. The helium nucleus
should therefore consist of 4 hydrogen nuclei, which
would together account for the atomic weight of 4.
But since these represent 4 positive charges, there must
also be present in the nucleus 2 negative electrons to
make the resultant nuclear charge equal to 2. We
could indeed hardly conceive of a system composed of
4 positive hydrogen nuclei alone ; for the forces of
repulsion would soon drive the separate parts asunder.
The two electrons can, so to speak, serve to hold the
system together. Fig. 24 gives a rough representation

THE NUCLEAR ATOM

of the helium atom. It must be carefully noted that
the picture is purely schematic and the distances
arbitrary. The helium nucleus, composed of 4 hydrogen
nuclei and 2 electrons, seems to possess extreme stability,
and it is not improbable that helium nuclei occur as
higher units in the structure of the nuclei of not only
the radioactive substances but also the other elements.
We shall perhaps be very near the truth in saying that
all nuclei are built up of combinations of hydrogen
nuclei, helium nuclei and electrons.

In nitrogen, with the atomic weight 14 and atomic

Fig. 24. — Schematic representation of a helium atom. K, nuclear
system with four hydrogen nuclei and two nuclear electrons ;
E, electrons in the outer electron system.

number 7, the nucleus should consist of 14 hydrogen
nuclei (with 12 of them compounded, perhaps, into 3
helium nuclei) and 7 nuclear electrons, reducing the
resultant positive nuclear charge from 14 to 7. Uranium, •
with atomic number 92 and atomic weight 238, should
have a nucleus composed of 238 hydrogen nuclei and
146 electrons, and so on for the others. We see at once
that the conception of the nucleus here propounded
leads us back to the old hypothesis of Prout (see p. 15)
that all atomic weights should be integral multiples of
that of hydrogen. This hypothesis apparently dis-
7

THE ATOM AND THE BOHR THEORY

agreed with atomic weight measurements, but the isotope
researches have vanquished this difficulty ; thus it has
been mentioned before that chlorine with an atomic
weight of 35*5 appears to be a mixture of isotopes
with atomic weights 35 and 37, and other cases have a
similar explanation. Yet the rule cannot be wholly
and completely exact. For, in the first place, the mass
of the electrons must contribute something, though this
contribution is far too small to be measured. But
there is also a second matter which plays a part here.
This is the law enunciated by Einstein in his relativity
theory, that every increase or decrease in the energy of a
body is correlated with an increase or decrease in the
mass of the body, proportional to the energy change.
We must, therefore, expect that the masses of the
various atomic nuclei will depend not only on the
number of hydrogen nuclei (and electrons), but also on
the energy represented in the attractions and repulsions
between the particles of the system, and in their mutual
motions, or the energy which comes into play in the
formation and disintegration of nuclear systems. This
is presumably closely connected, although in a way
which is not clearly understood, with the fact, that if
the atomic weights of the elements are to come out
integers, that of hydrogen must not be taken as 1 but
as 1*008 ; that is, the atomic weight unit must be
chosen a little smaller than the atomic weight of hydrogen
(cf. table, p. 23).

Transformation of Elements and Liberation of Atomic
Energy.

We shall now treat very briefly two questions which
have profoundly interested many people, because they

THE NUCLEAR ATOM

are concerned with possible practical applications of our
new knowledge of atoms.

The first question is this : Can one not, from this
knowledge, bring about the transformation of one
element into another ? In answering this, it can, of
course, be said immediately that among the radio¬
active substances such transformations are constantly
taking place without human interference, and we cer¬
tainly have no right to state offhand that it will be
impossible for man ever to bring about such a transfor¬
mation artificially. For example, if we could succeed in
getting one hydrogen nucleus loose from the nucleus of
mercury, the latter would thereby be changed into a
gold nucleus. Such a thing is not only conceivable, but
in the last few years it has become a reality, though,
to be sure, not with the substances here mentioned.
In 1919 Rutherford, by bombarding nitrogen (N = i4)
with ct-particles, was able to knock loose some hydro¬
gen nuclei from the nitrogen nucleus ; perhaps he
succeeded thereby in changing the nitrogen nuclei
into carbon nuclei (C = i2) by the breaking off of two
hydrogen nuclei from each nitrogen nucleus. But to
disintegrate very few nitrogen nuclei, Rutherford had
to employ a formidable bombardment with hundreds
of thousands of projectiles (a-particles) ; and even if he
had ended with gold instead of carbon, this would have
been, from the economic point of view, a very foolish way
of making gold ; and at the present time we know of no
other artificial method for the transformation of elements.
That Rutherford’s investigation has, in any case, extra¬
ordinarily great interest and scientific value is another
matter.

The second question is whether one cannot liberate

100 THE ATOM AND THE BOHR THEORY

and utilize the energy latent in the interior of the atom.
This question, which was suggested in the first instance
by the discovery of radium, has recently attracted con¬
siderable attention because of reports that, according
to Einstein’s relativity theory, one gram of any substance
by virtue of its mass alone must contain a quantity of
energy equal to that produced by the burning of 3000
tons of coal. The meaning of this statement is this :
it has already been mentioned that according to the
relativity theory a decrease in the energy of a body
brings about a decrease in its mass ] it is immaterial in
what form the energy is given up, whether as heat, elastic
oscillations, or the like ; all that is said is, that to a
certain decrease in mass, will correspond a perfectly
definite emission of energy in some form. If we now
could imagine the whole mass of one gram of a substance
to be “ destroyed ” (i.e. caused to disappear utterly as a
physical substance), and to reappear as heat energy, for
example, then we could compute from the known rela¬
tion between mass and energy, that the heat energy
thus brought about would be equivalent to that ob¬
tained by the burning of 3000 tons of coal. But in
order that all this energy should be developed, even
the hydrogen nuclei and the electrons would have to be
“ destroyed,” and no phenomenon is known, support¬
ing the supposition that such a “ destruction ” of the
fundamental particles of a substance is possible, or
that it is possible to transform these particles into
other types of energy. A thought like this must rather
be stamped as fantasy, the origin of which is to be
found in a misunderstanding of a purely scientific mode
of expression.

The case is essentially different with those quantities

THE NUCLEAR ATOM

101

of energy which must be assumed to be freed or absorbed
in the transformation of one nuclear system into another,
that is, in elemental transformations. Though these
are far smaller in amount, the radioactive processes
indicate that they are not wholly to be despised. For
one gram of radium will upon complete disintegration to
non-radioactive material give off as much energy as is
equivalent to 460 kg. of coal. But even here we must
confess that it will take about 1700 years for only half of
the radium to be transformed. It is not at all impossible
that other elemental transformations might lead to
just as great energy developments as appear in the
disintegration of radioactive substances. Let us imagine
that four hydrogen nuclei, which together have a mass of
4 = 1*008=4*032, and two electrons could join together
to form a helium nucleus with atomic weight very
close to 4. This process would thus result in a loss of
mass which must be assumed to appear in another form
of energy. The amount of energy obtainable in this way
from one gram of hydrogen would be considerably more
than that given off by the disintegration of one gram of
radium.

There can hardly exist any doubt that in nature there
occur not only disintegrations, but also (perhaps in the
interior of the stars) building-up processes in which
compound nuclei result from simple ones. It is therefore
natural to suppose that by exerting on hydrogen excep¬
tional conditions of temperature, pressure, electrical
changes, etc., we could succeed by experiments here on
earth in forming helium from it with the development
of considerable energy. But at the same time it is very
likely that even under favourable circumstances such a
process would take place with very great slowness,

102 THE ATOM AND THE BOHR THEORY

because the formation of a helium nucleus might well
be a very infrequent occurrence ; it would probably be
the result of a certain succession of collisions between
hydrogen nuclei and electrons, a combination whose prob¬
ability of occurrence in a certain number of collisions is
infinitely less than the probability of winning the largest
prize in a lottery with the same number of chances.
Nature has time enough to wait for “ wins,” while man¬
kind unfortunately has not. We know concerning the
disintegration of the radioactive substances that it is of
the character here indicated ; of the great number of
atoms to be found in a very small mass of a radioactive
substance, now one explodes and now another. But
why fortune should pick out one particular atom is as
difficult to understand as why in a lottery one particular
number should prove to be the lucky one rather than
any other. Our only understanding of the whole matter
rests on the law of averages, or probability as we may call
it. We know that of a billion radium atoms (io12) on
the average thirteen explode every second ; and even if
in any single collection of a billion a few more or a few
less may explode, the average of thirteen per second per
billion will always be maintained in dealing with larger
and larger numbers of atoms, as, for example, with a
thousand billion or a million billion. For other radio¬
active substances we get wholly different averages for
the number of atoms disintegrating per second ; but in
no case are we able to penetrate into the inner character
of the process of disintegration itself. And what holds
true of the radioactive substances will also hold true
probably for elemental changes of all kinds ; Rutherford
with his hundreds of thousands of a-particle projectiles
was able to make sure of but a few lucky " shots.” The

THE NUCLEAR ATOM

103

whole matter must at this stage be looked upon as

governed wholly by chance.

One interested in speculating on what would happen
if it were possible to bring about artificially a trans¬
formation of elements propagating itself from atom
to atom with the liberation of energy, would find
food for serious thought in the fact that the quantities
of energy which would be liberated in this way would
be many, many times greater than those which we
now know of in connection with chemical processes.
There is then offered the possibility of explosions more
extensive and more violent than any which the mind
can now conceive. The idea has been suggested that
the world catastrophes represented in the heavens by
the sudden appearance of very bright stars may be the
result of such a release of sub-atomic energy, brought
about perhaps by the “ super- wisdom of the unlucky
inhabitants themselves. But this is, of course, mere
fanciful conjecture.

It seems clear, however, that we need have no fear
that in investigating the problem of atomic energy
we are releasing forces which we cannot control, because
we can at present see no way to liberate the energy of
atomic nuclei beyond that which Nature herself provides,
to say nothing of a practical solution of the energy
problem. The time has certainly not yet come for the
technician to follow in the theoretical investigator’s
footsteps in this branch of science. One hesitates,
however, to predict what the future may bring
forth.

Interesting and significant as is the insight which
Rutherford and others have opened up into the inner
workings of the nucleus, the study of the electron

104 THE ATOM AND THE BOHR THEORY

system of the atom bears more intimately upon the
various branches of physical and chemical science,
and hence presents greater possibilities of attaining,
in a less remote future, to discoveries of practical
significance.

CHAPTER V

THE BOHR THEORY OF THE HYDROGEN

SPECTRUM

The Nuclear Atom and Electrodynamics.

Even if Rutherford had not yet succeeded in giving
a complete answer to the first of the questions pro¬
pounded in the previous chapter, namely, that concerning
the positions of the positive and negative particles of the
atom, one might at any rate hope that his general
explanation of the structure of the atom — that is, the
division into the nucleus and surrounding electrons,
and the determination of the number of electrons in
the atoms of the various elements — would furnish a
good foundation for the answer to the second question
about the connection between the atomic processes
and the physical and chemical properties of matter.
But in the beginning this seemed so far from being true
that it appeared almost hopeless to find a solution of
the problem of the atom in this way.

We shall best understand the meaning of this if we
consider the simplest of the elemental atoms, namely,
the atom of hydrogen with its positive nucleus and its
one electron revolving about the nucleus. How could
it be possible to explain from such a simple structure
the many sharp spectral lines given by the Balmer-

105

106 THE ATOM AND THE BOHR THEORY

Ritz formula (p. 57) ? As has previously been men¬
tioned, the classical electron theory seemed to demand
a very complicated atomic structure for the explanation
of these lines. According to the electron theory, the
atoms may be likened to stringed instruments which
are capable of emitting a great number of tones, and
in these atoms the electrons are naturally supposed to
correspond to the “ strings.” But the hydrogen atom
has only one electron, and it hardly seems credible
that in a mass of hydrogen the individual atoms would
be tuned for different “ tones,” with definite frequencies
of vibration.

Now, it certainly cannot be concluded from the
analogy with the stringed instrument that a single
electron can emit light of only a single frequency at
one time, corresponding to a single spectral line. For a
plucked string will, as we know, give rise to a simple
tone only if it vibrates in a very definite and particu¬
larly simple way ; in general it will emit a compound
sound which may be conceived as made up of a “ funda¬
mental ” and its so-called “ overtones,” or " harmonies ”
whose frequencies are 2, 3, . . . times that of the funda¬
mental ( i.e . integral multiples of the latter). These
overtones may arise even separately because the string,
instead of vibrating as a whole, may be divided into
2, 3, . . . equally long vibrating parts, giving respec¬
tively 2, 3, . . . times as great frequencies of vibration.
We call such vibrations “ harmonic oscillations.” The
simultaneous existence of these different modes of
oscillation of the string may be thought of in the same
way as the simultaneous existence of wave systems of
different wave-lengths on the surface of water. Corre¬
sponding to the possibility of resolving the motion

THEORY OF HYDROGEN SPECTRUM 107

of the string into its “ harmonic components,” the
compound sound waves produced by the string can
be resolved by resonators (cf. p. 44) into tones possessing
the frequencies of these components.

According to the laws of electrodynamics the situa¬
tion with the electron revolving about the hydrogen
nucleus might -be expected to be somewhat similar to
that described above in connection with the vibrating
string. If the orbit of the electron were a circle, it
should emit into the ether electro-magnetic waves of a
single definite wave-length and corresponding frequency,
v, equal to co, the frequency of rotation of the electron
in its orbit ; that is, the number of revolutions per
second. But just as a planet under the attraction of
the sun, varying inversely as the square of the distance,
moves in an ellipse with the sun at one focus, so the
electron, under the attraction of the positive nucleus,
which also follows the inverse square law, will in general
be able to move in an ellipse with the nucleus at one
focus. The electromagnetic waves which are emitted
from such a moving electron may on the electron theory
be considered as composed of light waves corresponding
to a series of harmonic oscillations with the frequencies :

vx=co, v2=2co, v3=3sy . . . and so on,

where co, as before, is the frequency of revolution of the
electron. According as the actual orbit deviates more
or less from a circle, the frequencies v2, ^3 • • •
appear stronger or weaker in the compound light waves
emitted. But the actual distribution of spectral lines
in the real hydrogen spectrum presents no likeness
whatever to this distribution of frequencies.

From this it is evident that no agreement can be

108 THE ATOM AND THE BOHR THEORY

reached between the classical electron theory on the one
hand and the Rutherford atom model on the other.
Indeed, the disagreement between the two is really far
more fundamental than has just been indicated. Ac¬
cording to Lorentz’s explanation of the emission of light
waves, the electrons in a substance (see again p. 75)
should have certain equilibrium positions, and should
oscillate about these when pushed out of them by some
external impulse. The energy which is given to the
electron by such an impulse is expended in the emission
of the light waves and is thus transformed into radiation
energy in the emitted light, while the electrons fall to
rest again unless they receive in the meantime a new
impulse. We can get an understanding of what these
impulses in various cases may be by thinking of them,
in the case of a glowing solid, for example, as due to the
collisions of the molecules ; or in the case of the glowing
gas in a discharge tube, from the collisions of electrons
and ions. The oscillating system represented by the
electron (the “ oscillator ”) will possess under these
circumstances great analogy with a string which after
being set into vibration by a stroke gradually comes back
to rest, while the energy expended in the stroke is emitted
in the form of sound waves. Although the vibrations
of the string become weaker after a while, the period of
the vibrations will remain unchanged ; the string
vibrations like pendulum oscillations have an invariable
period, and the same will be the case with the frequency
of the electron if the force which pulls it back into its
equilibrium position is directly proportional to the dis¬
placement from this position (the “ harmonic motion ”
force).

Rutherford’s atomic model is, however, a system of a

THEORY OF HYDROGEN SPECTRUM 109

kind wholly different from the “oscillator” of the
electron theory. The one revolving hydrogen electron
will find a position of “ rest ” or equilibrium only in the
nucleus itself, and if it once becomes united with the
latter it will not easily escape ; it will then probably
become a nuclear electron, and such a process would be
nothing less than a transformation of elements (see p. 79).
On the other hand, it follows necessarily from the funda¬
mental laws of electrodynamics that the revolving
electron must emit radiation energy, and, because of
the resultant loss of energy, must gradually shrink its
path and approach nearer the nucleus. But since the
nuclear attraction on the electron is inversely propor¬
tional to the square of the distance, the period of revolu¬
tion will be gradually decreased and hence the frequency
of revolution a, and the frequency of the emitted fight
will gradually increase. The spectral fines emitted from
a great number of atoms should, accordingly, be distri¬
buted evenly from the red end of the spectrum to the
violet, or in other words there should be no fine spectrum
at all. It is thus clear that Rutherford’s model was not
only unable to account for the number and distribution
of the spectral fines ; but that with the application of the
ordinary electrodynamic laws it was quite impossible
to account for the existence even of spectral fines.
Indeed, it had to be admitted that an electrodynamic
system of the kind indicated was mechanically unstable
and therefore an impossible system ; and this would
aPPly n°t merely to the hydrogen atom, but to all
nuclear atoms with positive nuclei and systems of
revolving electrons.

However one looks at the matter, there thus appears
to be an irremediable disagreement between the Ruther-

110 THE ATOM AND THE BOHR THEORY

ford theory of atomic structure and the fundamental
electrodynamic assumptions of Lorentz’s theory of
electrons. As has been emphasized, however, Ruther¬
ford founded his atomic model on such a direct and clear-
cut investigation that any other interpretation of his
experiments is hardly possible. If the result to which
he attained could not be reconciled with the theory of
electrodynamics, then, as has been said, this was so
much the worse for the theory.

It could, however, hardly be expected that physicists
in general would be very willing to give up the conceptions
of electrodynamics, even if its basis was being seriously
damaged by Rutherford’s atomic projectiles. Sur¬
mounted by its crowning glory — the Lorentz electron
theory — the classical electrodynamics stood at the
beginning of the present century a structure both solid
and spacious, uniting in its construction nearly all the
physical knowledge accumulated during the centuries,
optics as well as electricity, thermodynamics as well as
mechanics. With the collapse of such a structure one
might well feel that physics had suddenly become home¬
less.

The Quantum Theory.

In a field completely different from the above the ^
conclusion had also been reached that there was some- *
thing wrong with the classical electrodynamics.
Through his very extended speculations on thermo¬
dynamic equilibrium in the radiation process, Planck
(1900) had reached the point of view expressed in his
quantum theory, which was just as irreconcilable with the
fundamental electro dynamic laws as the Rutherford
atom.

THEORY OF HYDROGEN SPECTRUM 111

A complete representation of this theory would lead
us too far ; we shall merely give a short account of the
foundations on which it rests.

By a black body is generally understood a body which
absorbs all the light falling upon it, and, accordingly, can
reflect none. Physicists, however, denote by the term
“ perfect black body ” in an extended sense, a body which
at all temperatures absorbs all the radiation falling upon
it, whether this be in the form of visible light, or ultra¬
violet, or infra-red radiation. From considerations which
were developed some sixty years ago by Kirchhoff, it
can be stated that the radiation which is emitted by such
a body when heated does not depend on the nature of
the body but merely on its temperature, and that it is
greater than that emitted by any other body whatever
at the same temperature. Such radiation is called tem¬
perature radiation or sometimes “ black ” radiation,
though the latter term is apt to be misleading, since a
" perfect black body ” emitting black radiation may
glow at white heat. It may be of interest to note here
the fundamental law deduced by Kirchhoff, which may
best be illustrated by saying that good absorbers of
radiation are also good radiators. An instructive experi¬
ment illustrating this is performed by painting a figure in
lampblack on a piece of white porcelain. The lampblack
surface is clearly a better absorber of radiant energy
than the white porcelain. When the whole is heated in
a blast flame, the lampblack figure glows much more
brightly than the surrounding porcelain, thus showing
that at the same temperature it is also the better radiator.
Following the same law we conclude that highly reflecting
bodies are not good radiators, a fact that has practical
significance in house heating. The perfect black body,

112 THE ATOM AND THE BOHR THEORY

then, being the best absorber of radiation, is also the
best radiator.

In actual practice no body is absolutely black. Even
a body coated with lampblack reflects about io per cent,
of the light waves incident on it. The Danish physicist
Christiansen remarked long ago that a real black body
could be produced only if an arrangement could be made
whereby the incident waves could be reflected several
times in succession before finally being emitted. To take
the case of lampblack, three such reflections would reduce
the re-emission from the lampblacked body to only i / 1000
of the radiation initially falling on it. This type of
black body was finally realized by making a cavity in an
oven having as its only opening a very small peep-hole,
and keeping the temperature of the wall of the oven
uniform. If a ray is sent into the cavity through the
peep-hole, it will become, so to speak, captured, because,
when once inside it will suffer countless reflections from
the walls of the cavity, having more and more of its
energy absorbed at each reflection. Very little of the
radiation thus entering will ever get out again, and con¬
sequently such a body will act very much like a “ perfect
black body ” according to the theoretical definition above.
Accordingly, the radiation which is emitted from such a
glowing cavity through the peep-hole will be black or
practically black radiation.

The cavity itself will be criss-crossed in all directions
by radiation emitted from one part of the inner surface
of the cavity and absorbed by (and partially reflected
from) other parts of the surface. When the walls of
the cavity are kept at a fixed and uniform tempera¬
ture there will automatically be produced a state
of equilibrium in which every cubic centimetre of the

THEORY OF HYDROGEN SPECTRUM 113

cavity will contain a definite quantity of radiation
energy, dependent only on the temperature of the walls.
Further, in the equilibrium state the radiation energy will
be distributed in a perfectly definite way (dependent as
before on the temperature only) among the various types
of radiation corresponding to different wave-lengths and
frequencies. If there is too much radiation of one kind
and too little of another, the walls of the cavity will
absorb more of the first kind than they emit, and emit
more of the second kind than they absorb, and so the
state will vary until the right proportion for equilibrium
is attained.

This distribution of the radiation over widely differing
wave-lengths can be investigated by examining spectro¬
scopically the light emitted from the peep-hole in the
cavity. Then, by means of a bolometer or some other
instrument, the heat development in the different portions
of the spectrum can be measured. At a temperature of
1500° C., for example, one will find that the maximum
energy is represented for rays of wave-length in the close
vicinity of i-8 fjb, i.e. in the extreme infra-red. If the
temperature is raised, the energy maximum travels off
in the direction of the violet end of the spectrum ; if the
temperature is lowered, it will move farther down into the
infra-red.

It is also possible to make a theoretical calculation
of the distribution of energy in the spectrum of the black-
body radiation at a given temperature. But the results
obtained do not agree with experiment. The English
physicists, Rayleigh and Jeans, developed on the basis
of the classical electro-dynamic laws and by apparently
convincing arguments a distribution law according to
which actual radiation equilibrium becomes impossible,
8

114 THE ATOM AND THE BOHR THEORY

since if it were true the energy in the radiation would
tend more and more to go over to the region of short
wave-lengths and high frequencies, and this shifting
would apparently go on indefinitely. The theory thus
leads to results which are not only in disagreement with
experiment, but which must be looked upon as extremely
unreasonable in themselves.

Planck, however, had vanquished these difficulties
and had obtained a radiation law in agreement with
experiment by introducing an extremely curious hypo¬
thesis. Like Lorentz, he thought of radiation as pro¬
duced through the medium of small vibrating systems or
oscillators, which could emit or absorb rays of a definite
frequency v. But while, according to the Lorentz theory
and the classical electrodynamics, radiation can be
emitted in infinitely small quantities ( i.e . small without
limit), Planck assumed that an oscillator can emit and
absorb energy only in certain definite quantities called
quanta , where the fundamental quantum of radiation is
dependent on the frequency of the oscillator, varying
directly with the latter. If thus we denote the smallest
quantity of energy which an oscillator of frequency
v can emit or absorb by E, then we can write

~E=hv,

where h is a definite constant fixed for all frequencies.
Accordingly the cavity can receive radiation energy of
frequency v from the radiating oscillators in its wall, or
transfer energy to these in no smaller quantity than hv.
The total energy of that kind emitted or absorbed at
any given time will always be an integral multiple of hv.
Oscillators with a frequency ij times as great will emit
energy in quanta which are times larger, and so on.

THEORY OF HYDROGEN SPECTRUM 115

The quantity h is independent, not only of the wave¬
length, but also of the temperature and nature of the
emitting body. This constant, the so-called Planck
constant , is thus a universal constant. If one uses the
“ absolute ” units of length, mass and time (see table,
p. 210), its value comes out as 6*54 xio-27. For the
frequency 750 X io12 vibrations per second, corresponding
to the extreme violet in the visible spectrum, the Planck
energy quantum thus becomes about 5 X io-12 erg, or
3-69 Xio-19 foot-pounds (note that the “ erg ” is the
“ absolute ” unit of work, or the amount of work done
when a body is moved through a distance of 1 cm. by a
force of one dyne acting in the direction of the motion,
while the “ foot-pound ” is the work done when a force
of 1 pound moves a body 1 foot in the direction of the
force). For light belonging to the red end of the
spectrum, the energy is about half as great. If we pass,
however, to the highest frequencies and the shortest
wave-lengths which are known, namely those corre¬
sponding to the “ hardest ” (i.e. most penetrating)
y-rays (see p. 78), we meet with energy quanta which
are a million times larger, i.e. 2 Xio~6 erg, although they
are still very small compared to any amount of energy
measurable mechanically.

This remarkable theory of quanta, which in the
hands of Planck still possessed a rather abstract character,
proved under Einstein’s ingenious treatment to have the
greatest significance in many problems which, like heat
radiation, had provided physicists with many difficulties.
For by assuming that energy in general could only be
given up and taken in in quanta, certain facts about the
specific heats of bodies could be accounted for — facts
which the older physics had proved powerless to explain.

116 THE ATOM AND THE BOHR THEORY

The Planck energy quanta, as Einstein showed, could also
explain in a very direct and satisfactory way the photo¬
electric effect , as it is called. This effect consists in the
freeing of electrons from a metal plate which ultra¬
violet rays are allowed to strike. The maximum velocity
with which these electrons are propelled from the plate
is found to be independent of the intensity of the incident
light, but dependent simply on the frequency of the radia¬
tion. Careful measurements have indeed shown, as
Einstein predicted, that the incident light really does
utilize an energy quantum hv to free each electron and
give it velocity (cf. p. 172). Of the different methods
which nowadays are at hand, the photoelectric effect
constitutes one of the best means of determining the value
of h. It has been applied for that purpose by Millikan,
to whose ingenious experiments the most accurate direct
determination of h is actually due.

All this lay completely outside the laws of electro¬
dynamics, and pointed to the existence of unknown
and more fundamental laws. But, for the time being,
physicists had to be satisfied merely with a recognition
of the fact that these mysterious energy quanta play a
very significant part in many phenomena.

A decade ago physics, as regards radiation problems,
was in a very unsettled state ; with four separate
branches of knowledge, each of which seemed firm and
well-founded enough in itself, but which had no common
connecting fink, indeed, were even to some extent
inconsistent with each other. The first of these was
the classical electrodynamics surmounted by the
felicitous electron theory of Lorentz and Larmor. The
second was the empirical knowledge of the spectra
resting on the work of Balmer, Ritz and Rydberg.

THEORY OF HYDROGEN SPECTRUM 117

The third was Rutherford’s nuclear atom model. And
the fourth was Planck’s quantum theory of heat radia¬
tion. It was quite evident that progress in the theory
of radiation and the structure of the atom was hope¬
less as long as these four points of view remained un-
correlated.

Main Outlines of the Bohr Theory.

Such was the situation when, in 1913, Bohr published
his atomic theory, in which he was able with great
ingenuity to unite the nuclear atom, the Balmer-Ritz
formula and the quantum theory. As far as electro¬
dynamics is concerned, the impossibility of retaining
that in its classical form was presented in a much clearer
way than ever before. But, as will presently be evident,
the Bohr theory has a very definite connection with
the classical theory, and Bohr’s attempts to preserve
and develop this connection have proved to be of the
greatest significance for his theory. In spite of the
fundamental rupture with the old ideas, the Bohr
theory strives to absorb all that is useful in the classical
point of view.

At the head of the theory appear the two funda¬
mental hypotheses or postulates on the properties of
the atom.

The first postulate states that for each atom or atomic
system there exists a number of definite states of motion ,
called “ stationary states,” in which the atom {or atomic
system) can exist without radiating energy. A finite
change in the energy content of the atom can take place
only in a process in which the atom passes completely
from one stationary state to another.

The second postulate states that if such a transition

118 THE ATOM AND THE BOHR THEORY

takes place with the emission or absorption of electro¬
magnetic light waves, these waves will have a definite
frequency, the magnitude of which is determined by the
change in the energy content of the atom. If we denote
the change in energy by E and the frequency by v we may
write

E =hv, or y=5
h

where h is the Planck constant. In consequence of the
second postulate the emission as well as the absorption of
energy by the atom always takes place in quanta.

The two postulates say nothing concerning the nature
of the motion in the stationary states. In the applica¬
tions, however, a connection with the Rutherford
atomic model is established. Confining our attention
first to the hydrogen atom, the system with which we
are concerned consists, accordingly, of a positive nucleus
and one electron revolving about it. The various states
of motion which the electron can assume in virtue of
the first postulate are a series of orbits at different
distances from the nucleus. In each of these “ stationary
orbits ” the electron follows the general mechanical laws
of motion ; i.e. under the nuclear attraction which is
inversely proportional to the square of the distance,
the electron describes an ellipse with the nucleus at
one focus, as has previously been stated ; but in contradic¬
tion with the classical electrodynamics it will emit no
radiation while moving in this orbit. Fig. 25 shows a
series of these orbits, to which the numbers 1, 2, 3, 4
have been attached, and which for simplicity are repre¬
sented as circular.

If the electron passes from an outer orbit to an inner
one ; for example, if it goes from number 4 to number 2,

THEORY OF HYDROGEN SPECTRUM 119

or from number 2 to number I, the electric force which
attracts it to the nucleus will do work just as the force
of gravity does work when a stone falls to the ground.

A part of this work is used to increase the kinetic energy
of the electron, making its velocity in the inner orbit
greater than in the outer, but the rest of the work
is transformed into radiation energy which is emitted
from the atom in the form of monochromatic light. 1

Fig. 25. — The Bohr model of the hydrogen
atom in the simplified form (with
circles instead of ellipses) .

In consequence of the second postulate the frequency
of the emitted radiation is proportional to the energy
loss. When the electron has reached the innermost
orbit (the one denoted by 1 in the figure), it cannot get
any nearer the nucleus and hence cannot emit any more
radiation unless it first is impelled to pass from its inner
orbit to an outer orbit again by the absorption of external
energy sufficient to bring about this change. Once in
the outer orbit again, it is in a state to produce radiation
by falling in a second time. The innermost orbit

120 THE ATOM AND THE BOHR THEORY

represents thus the electron's equilibrium state, and
corresponds to the normal state of the atom.

If we try to illustrate the matter with an analogy
from the theory of sound, we can do so by comparing
the atom not with a stringed instrument, but with a
hypothetical musical instrument of a wholly different
kind. Let us imagine that we have placed one over
another and concentrically a series of circular discs of
progressively smaller radii, and let us suppose 'that a
small sphere can move around any one of these without
friction and without emitting sound. In such a motion
the system may be said to be in a “ stationary state."
Sooner or later the sphere may fall from the first disc
on to one lower down and continue to roll around on
the second, having emitted a sound, let us assume, by
its fall. By passing thus from one stationary state to
another it loses a quantity of energy equal to the work
which would be necessary to raise it again to the disc
previously occupied, and to bring it back to the original
state of motion. We can assume that the energy which
is lost in the fall reappears in a sound wave emitted by
the instrument, and that the pitch of the sound emitted
is proportional to the energy sent out. If, moreover,
we imagine that the lowermost disc is grooved in
such a way that the sphere cannot fall farther,
then this fanciful instrument can provide a very
rough analogy with the Bohr atom. We must beware,
however, of stretching the analogy farther than is
here indicated.

It must be specially emphasized here that the fre¬
quency of the sound emitted in the above example has
no connection with the frequency of revolution of the
sphere. In the Bohr atom, likewise, the frequency of

THEORY OF HYDROGEN SPECTRUM 121

revolution oo of the electron in its stationary orbit has
no direct connection with the frequency of the radiation
emitted when the electron passes from this orbit to
another. This is a very surprising break with all
previous views on radiation, a break whose revolutionary
character should not be under-estimated. But, however
unreasonable it might seem to give up the direct con¬
nection between the revolutional frequency and the
radiation frequency, it was absolutely necessary if the
Rutherford atomic model was to be preserved. And
as we shall now see, the new point of view of the Bohr
theory leads naturally to an interpretation of the Balmer-
Ritz formula, which had previously not been connected
with any other physical theory.

The quantity of energy E, which the atom gives up
when the electron passes from an outer to an inner
orbit, or which, conversely, is taken in when the electron
passes from an inner to an outer orbit, may, as has been
indicated, be regarded as the difference between the
energy contents of the atom in the two stationary
states. This difference may be expressed in the follow¬
ing way. Let us imagine that we eject the electron
from a given orbit (e.g. No. 2 in the diagram) so that it
is sent to “ infinity,” or, in other words, is sent so far
away from the nucleus that the attraction of the latter
becomes negligible. To bring about this removal of
the electron from the atom demands a certain amount
of energy, which we can call the ionizing work corre¬
sponding to the stationary orbit in question. We
may here designate it as A2. To eject the electron
from the orbit No. 4 will demand a smaller amount of
ionizing work, A4. The difference A2-A4 is accordingly
the work which must be done to transfer the electron

122 THE ATOM AND THE BOHR THEORY

from the orbit No. 2 to the orbit No. 4. This is, how¬
ever, exactly equal to the quantity E of energy which
will be emitted as light when the electron passes from
orbit No. 4 to orbit No. 2. If we call the frequency of
this light v, then from the relations E —hv and E =
A2— A4, we have

hv = A2 — A4

If, now, in place of this specific example using the
stationary orbits 2 and 4 we take any two orbits desig¬
nated by the numbers n" (for the inner) and n' (for the
outer), we can write for the frequency of the radiation
emitted for a transition between these arbitrary states

hv= An — An> or

h h

We have now reached the point where we ought to bring
in the Balmer-Ritz formula for the distribution of the
lines in the hydrogen spectrum. This formula may be
written (see p. 59)

_ K K
' w"2 «'2

We can now see very clearly the similarity between the
formula derived from the spectrum investigations and
that derived from the two Bohr postulates. In both
formulae the frequency appears as the difference between
two terms which are characterized in both cases by two
integral numbers, in the first formula, numbers denoting
two stationary orbits in the Bohr model for hydrogen,
and in the second the two numbers which in the Balmer-
Ritz formula for the hydrogen spectrum characterize,
respectively, a series and one of the lines of the series.
To obtain complete agreement we have merely to equate

THEORY OF HYDROGEN SPECTRUM 123

the corresponding terms in the two formulae. Thus
we have for any arbitrary integer n

4?=Iror A»=

h n 2

For the innermost stationary orbit, for which n — I,
the ionizing work A1 will accordingly be equal to the
product of the constants h and K of Planck and Balmer
respectively; and for the orbits No. 2, No. 3, No. 4, etc.,
the values will be respectively 1/4, 1/9, 1/16, etc., of
this product. From the charges on the nucleus and
the electron, which are both equal to the elementary
quantum e of electricity (see p. 90), and from the ionizing
energy for a given orbit we can now find by the use of
simple mechanical considerations the radius of the orbit.
If we denote the radii of the orbits 1, 2, 3 ... by av a2,
a3 . . ., we then obtain for the diameters 2 av 2 a2, 2 a3 . . .
the values 2% = 1*056 xio-8 cm. (or approximately
2<21 = io~8 cm.), 2a2 =4 xio-8 cm., 2#3=9Xio~8 cm., etc.
It is seen that the radii of the orbits are in the proportion
1, 4, 9 . . ., or in other words the squares of the integers
which determine the orbit numbers. It is in this propor¬
tion that the circles in Fig. 25 are drawn. We must
remember, however, that we have here for the moment
been thinking of the orbits as circles, while in reality
they must in general be assumed to be ellipses. The
foregoing considerations will, however, still hold with
the single change that 2 an will now mean, instead
of the diameter of a circle, the major axis of an
ellipse.

Let us return to the formulae

v = - —

h h

and

K _K
n"2 n'2

124 THE ATOM AND THE BOHR THEORY

Here n" denotes in the first formula the index number for
the inner of the two orbits between which the transition
is supposed to take place, while in the second formula n"
denotes a definite series in the hydrogen spectrum.
If n" is 2 while n' takes on the values 3, 4, 5 ... oo
then in the Bohr model of the hydrogen atom this
corresponds to a series of transitions to the orbit No. 2
from the orbits 3, 4, 5 • • •> while in the hydrogen
spectrum this corresponds to the lines in the Balmer
series, namely, the red line (Ha) corresponding to the
transition 3-2, the blue-green line (H/3) to 4-2, the violet
line (Hy) to 5-2 and so on. If we now put n" = 1 while
n' takes the values 2, 3, 4 . . ., we get in the atom transi¬
tions to the orbit No. 1 from the orbits No. 2, 3, 4 . . .,
corresponding in the spectrum to what is called the
Lyman series in the ultra-violet (named after the
American physicist Lyman, who has carried on extensive
researches in the ultra-violet region of the spectrum).
Thus every line in the hydrogen spectrum is represented
by a transition between two definite stationary states
in the hydrogen atom, since this transition will give
the frequency corresponding to the line in question.

At first sight this would seem perhaps to be such an
extraordinary satisfactory result that it would prove
an overwhelming witness in favour of the Bohr theory.
A little more careful thought, on the other hand, would
perhaps cause a complete reversion from enthusiasm
and lead some to say that the whole thing has not the
slightest value, because the stationary states were so
chosen that agreement might be made with the Balmer-
Ritz formula. This last consideration, indeed, states
the truth in so far that the agreement between the

THEORY OF HYDROGEN SPECTRUM 125

formula and the theory, at least as developed here up
to this point, is of a purely formal nature. In the
Bohr postulates the frequencies of the emitted radiation
are determined by a difference between two of a series
of energy quantities, characterizing the stationary states,
just as in the Balmer-Ritz formula they appear as a
difference between two of a series of terms (K, K/4,
K/9, . . .) each characterized by its integer. Now by
characterizing the quantities of energy in the stationary
states by a series of integers (in itself a wholly arbitrary
procedure) complete agreement between the Bohr
stationary states idea and the spectral formulae can be
attained. It is not even necessary to introduce the
Rutherford atomic model to attain this end. By
bringing in this specific model, one might join the new
theory to the knowledge already gained of the atomic
structure, and, so to speak, crystallize the hitherto
undefined or only vaguely defined stationary states
into more definite form as revolution in certain concrete
orbits. This would then lead to a more comprehensive
conception of atomic structure. But the theory un¬
fortunately would still be rather arbitrary, since there
would seem to be no justification for picking out certain
fixed orbits with definite diameters or major axes to
play a special role. One cannot wonder then that
many scientists considered the Bohr theory unaccept¬
able, or at any rate were inclined to look upon it simply
as an arbitrary, unreasonable conception which really
explained nothing.

Naturally, Bohr himself clearly recognized the formal
nature of the agreement between the Balmer-Ritz
formula and his postulates. But Bohr was the first to
see that the quantum theory afforded the possibility of

126 THE ATOM AND THE BOHR THEORY

bringing about such an agreement, and he saw, moreover,
that the agreement was not merely fortuitous, but con¬
tained within it something really fundamental, on which
one could build further. That atomic processes on his
theory took on an unreasonable character (compared with
the classical theory) was nothing to worry about, for
Bohr had come to the clear recognition that it was
completely impossible to understand from known laws
the Planck-Einstein “ quantum radiation,” or to deduce
the properties of the spectrum from the Rutherford atom
alone. He therefore saw that his theory was really
not introducing new improbabilities, but was only causing
the fundamental nature of the contradictions which had
previously hindered development in this field to appear
in a clearer light.

But in addition to this the choice of the dimensions
of the stationary states was by no means so arbitrary
as might appear in the foregoing. In his first presenta¬
tion of the theory of the hydrogen spectrum, Bohr had
derived his results from certain considerations connected
with the quantum theory — considerations of a purely
formal nature, indeed, just as those developed in the
preceding, but leading to agreement with the spectral
formulae. He, moreover, called attention to the fact
that the values obtained for the orbital dimensions were
of the same order of magnitude as those which could be
expected on wholly different grounds. The diameter
of the innermost orbit, i.e.} that which defines the outer
limit of the atom in the normal state, was found to be,
as has been noted above, about io-8 cm., i.e., of the same
order of magnitude as the values obtained for the
diameters of molecules on the kinetic theory of gases
(see p. 27). The stationary states corresponding to

THEORY OF HYDROGEN SPECTRUM 127

very high quantum numbers one could expect to meet
only when hydrogen was very attenuated, for otherwise
there could be no room for the large orbits. We note
that the 32nd orbit must have a diameter 32s (or over
1000 times) as great as the innermost orbit. Since,
now, lines with high number in a hydrogen series corre¬
spond on the Bohr theory to transitions from orbits of
high number to an inner orbit, it became understand¬
able why only comparatively few lines of the Balmer
series are ordinarily observed in the discharge tube,
while many more lines are observed in the spectra of
certain stars. For in such stars the possibility is left
open for hydrogen to exist in a very attenuated state,
and yet in such large masses that the lines in question
can become strong enough for observation. In fact,
one must assume that in a great mass of hydrogen a very
large number of atoms send out simultaneously light
of the wave-length corresponding to one line. For the
ionizing work, i.e., the work necessary to eject the
electron completely from the normal state and thus make
the atom into a positive ion, the Bohr theory gives a
value of the same order of magnitude as the so-called
“ ionization potentials ” which have been found by
experiment for various gases. An exact correspondence
between theory and experiment could for hydrogen not
be attained with certainty, because the hydrogen atoms
in hydrogen gas under ordinary conditions always
appear united in molecules.

In his very first paper, however, Bohr had studied
Balmer’s formula also from another point of view,
and had derived in this way an expression for the
Rydberg constant K which agreed with experiment.
These considerations have reference to the above

128 THE ATOM AND THE BOHR THEORY

mentioned connection of the theory with the classical
theory of electrodynamics.

Such a connection had previously been known to
exist in the fact that, for long wave-lengths, the radiation
formula of Planck reduces practically to the Rayleigh
Jeans Law which can be derived from electrodynamics.
This is related to the fact that when v is small (long
wave-lengths), the energy quantum hv is very small, and
hence the character of the radiation emitted will approach
more and more nearly to a continuous “ un quantized ”
radiation. One might then expect that the Bohr theory
also should lead in the limit of long wave-lengths and
small frequencies to results resembling those of the
ordinary electrodynamic theory of the radiation process.
On the Bohr theory we get the long wave-lengths for
transitions between two stationary states of high num¬
bers (numbers which also differ little from each other).
Thus suppose n is a very large number. Then the
transition from the orbit n to the orbit n — I will give rise
to radiation of great wave-length. For in this case An
and An_x differ very little, and accordingly hv is very
small, as must v be also. According to the electro¬
dynamic theory of radiation, the revolving electron should
emit radiation whose frequency is equal to the electron’s
frequency of revolution. According to the Bohr theory
it is impossible to fulfil this condition exactly, since
radiation results from a transition between two stationary
orbits in each of which the electron has a distinct re-
volutional frequency. But if n is a large number, the
difference between the frequencies of revolution con and
a)n— i for the two orbits n and n — i, respectively, becomes
very small ; for example, for n = ioo, it is only 3 per cent.
For a certain high value of n, then, the frequency of the

THEORY OF HYDROGEN SPECTRUM

129

emitted radiation can therefore be approximately equal
to the frequency of revolution of the electron in both the
two orbits, between which the transition takes place.
But even if this proved correct for values of n about ioo,
one could not be sure beforehand whether it would work
out right for still larger values of n, for example, 1000.

In order to investigate this latter point we must look
into the formulae for the revolutional frequency co in a
stationary orbit and for the radiation frequency v.
Since, according to the Bohr theory, we can apply the
usual laws of mechanics to revolution in a stationary
orbit, it is an easy matter to find an expression for a.
From a short mathematical calculation we can deduce
that <y=R/w3, where R is the frequency of revolution
for the first orbit (n — i). We find v, on the other hand,
by substituting in the Balmer-Ritz formula the numbers
n and n — I, and a simple calculation shows that for
great values of n, the expression for v will approach in
the limit the simple form v=2K/n3. For large orbit
numbers, v accordingly varies as u, i.e., inversely pro¬
portional to the third power of n , and by equating
R and 2K, we find that the values for v and 00 tend
more and more to become equal.

In this way the value of K, the Balmer constant, may
be computed. It is found that

K =2 7r2e*mlh3

where e is the charge on the electron, m the mass of the
electron, and h is Planck’s constant. Upon the sub¬
stitution of the experimental values for these quantities,
a value of K is determined which agrees with the experi¬
mental value (from the spectral lines investigation) of
3*29 Xio15 within the accuracy to which e, m and h are
9

130 THE ATOM AND THE BOHR THEORY

obtainable. This agreement has from the very first been
a significant support for the Bohr theory.

One might now object that we have here considered
radiation due to a transition between two successive
stationary states, e.g., No. ioo and No. 99, or the like
(a “ single jump ” we might call it). On the other hand,
for transitions between states whose numbers differ by
2, 3, 4 or more (as in a double jump, or a triple jump)
the agreement found above will wholly disappear, and
doubt be cast on its value. For in such cases of high
orbit numbers the frequency of revolution will remain
approximately the same even for a difference of 2, 3, 4 or
more in orbit number ; but the radiation frequency for
a double jump will be nearly twice that for a single
jump, while that for a triple jump will be nearly three
times, etc. Accordingly, for approximately the same
revolutional frequency co we shall have in these cases for
the radiation frequency very nearly v1=a, v2=2co, =

3 co, etc. We must, however, remember that when the
orbit in the stationary states is not a circle, but an
ellipse (as must in general be assumed to be the case),
the classical electrodynamics require that the electron
emits besides the “ fundamental ” radiation of frequency
iq =u) the overtones of frequencies v2=2gj, ^=3 00 . . .
We then also here see the outward similarity between
the Bohr theory and the classical electrodynamics. We
may say that the radiation of frequency v, produced by
a single jump, corresponds to the fundamental harmonic
component in the motion of the electron, while the
radiation of frequency v2, emitted by a double jump,
corresponds to the first overtone, etc.

The similarity is, however, only of a formal nature,
since the processes of radiation, according to the Bohr

THEORY OF HYDROGEN SPECTRUM 131

theory, are of quite different nature than would be
expected from the laws of electrodynamics. In order to
show how fundamental is the difference, even where the
similarity seems greatest, let us assume that we have a
mass of hydrogen with a very large number of atoms
in orbits, corresponding to very high numbers, and
that the revolutional frequency can practically be set
equal to the same quantity co. There may take
place transitions between orbits with the difference
i, 2, 3 ... in number, and as the result of these different
transitions we shall find, by spectrum analysis, in the
emitted radiation frequencies which are practically a,
2o), 3 a, etc. According to the radiation theory of electro¬
dynamics we should also get these frequencies and the
spectral lines corresponding to them. It must, however,
be assumed that they are produced by the simultaneous
emission from every individual radiating atom of a
fundamental and a series of overtones. According to
the Bohr theory, on the other hand, each individual
radiating atom at a given time emits only one definite
line corresponding to a definite frequency (monochromatic
radiation).

We can now realize that the Bohr theory takes us into
unknown regions, that it points towards fundamental
laws of nature about which we previously had no ideas.
The fundamental postulates of electrodynamics, which
for a long time seemed to be the fundamental laws of the
physical world itself, by which there was hope of explain¬
ing the laws of mechanics and of light and of everything
else, were disclosed by the Bohr theory as merely super¬
ficial and only applicable to large-scale phenomena.
The apparently exact account of the activities of nature,
obtained by the formulae of electrodynamics, often veiled

132 THE ATOM AND THE BOHR THEORY

processes of a nature entirely different from those the
formulae were supposed to describe.

One might then express some surprise that the laws
of electrodynamics could have been obtained at all and
interpreted as the most fundamental of all laws. It
must, however, be remembered that the Bohr theory for
large wave-lengths, i.e., the slow oscillations, leads to
a formal agreement with electrodynamics. It must,
moreover, be remembered that the laws of electro¬
dynamics are established on the basis of large-scale
electric and magnetic processes which do not refer to
the activities of separate atoms, but in which very
great numbers of electrons are carried in a certain
direction in the electric conductors or vibrate in oscilla¬
tions which are extremely sIqw compared with light
oscillations. Moreover, the observed laws, even if they
can account for many phenomena in light, early showed
their inability to explain the nature of the spectrum
and many other problems connected with the detailed
structure of matter. Indeed the more this structure was
studied, the greater became the difficulties, the stronger
the evidence that the solution cannot be obtained in the
classical way.

If we ask whether Bohr has succeeded in setting up
new fundamental laws, which can be quantitatively
formulated, to replace the laws of electrodynamics and
to be used in the derivation of everything that happens in
the atom and so in all nature, this question must receive
a negative answer. The motion of the electron in a given
stationary state may, at any rate to a considerable extent,
be calculated by the laws of mechanics. We do not know,
however, why certain orbits are, in this way, preferred
over others, nor why the electrons jump from outer to

THEORY OF HYDROGEN SPECTRUM 133

inner orbits, nor why they sometimes go from one
stationary orbit to the next and sometimes jump over
one or more orbits, nor why they cannot come any closer
to the nucleus than the innermost orbit, nor why, in
these transitions, they emit radiation of a frequency
determined according to the rules mentioned.

It must not be forgotten that in science we must
always be patient with the question “ Why ? ” We
can never get to the bottom of things. On account of
the nature of the problem, answers cannot be given to
the questions why the smallest material particles (for
the time being hydrogen nuclei and electrons) — the
elementary physical individuals — exist, or why the funda¬
mental laws for their mutual relationships — the most
elementary relationships existing between them — are
of this or that nature ; a satisfactory answer would
necessarily refer to something even more elementary.
We cannot claim more than a complete description of
the relative positions and motions of the fundamental
particles and of the laws governing their mutual action
and their interplay with the ether.

If we examine our knowledge of the atomic processes
in the light of this ideal we are tempted, however, to
consider it as boundless ignorance. We are incon¬
ceivably far from being able to give a description of the
atomic mechanism, such as would enable us to follow,
for example, an electron from place to place during its
entire motion, or to consider the stationary states as
links in the whole instead of isolated “ gifts from above.”
During the transition from one stationary state to another
we have no knowledge at all of the existence of the
electron, indeed we do not even know whether it exists
at that time or whether it perhaps is dissolved in the

134 THE ATOM AND THE BOHR THEORY

ether to be re-formed in a new stationary state. But
even if we turn aside from such a paradoxical considera¬
tion, it must be recognized that we do not know what
path the electron follows between two stationary orbits
nor how long a time the transition takes. As has been
done in this book, the transition is often denoted as a
jump, and many are inclined to believe that the electron
in its entire journey from a distant outer orbit to the
innermost spends the greatest part of the time in the
stationary orbits, while each transition takes but an
infinitesimally short time. This, however, in itself does
not follow from the theory, nor is it implied in the expres¬
sion “ the stationary states.” These states may in a
certain sense be considered as way stations ; but when
we ask whether an electron stays long in the station,
or whether the stationary state is simply a transfer point
where the electron changes its method of travelling so
that the frequency of its radiation is changed, these are
other matters, and we cannot here go into the considera¬
tions connected with them.

To get an idea of some of the difficulties inherent
in the attempt to make concrete pictures of the nature
of the processes, let us again consider the analogy
between the Bohr atom of hydrogen and a special kind
of musical instrument in which sounds are produced
by the fall of a small sphere between discs at various
heights (see p. 120). It will be most natural here to
think of the sounds as developed by the sphere when
it hits the lower disc, and to think of the tones of higher
pitch as given by the harder blows, corresponding to
the larger energy (determinative of the pitch) released
by the fall. We can, however, by no means transfer
such a picture to the atomic model. For in the latter

THEORY OF HYDROGEN SPECTRUM 135

we cannot think of the stationary state as a material
thing which the electron can hit, and it is also unreason¬
able to imagine that the radiation is not emitted until
the moment when the transition is over and the electron
has arrived in its new stationary state. We must, on
the contrary, assume that the emission of radiation
takes place during the whole transition, whether the
latter consumes a shorter or longer time. If it were
the case that a transition always took place between
two successive stationary states, it would then be
possible to use the musical instrument to illustrate the
matter. Let us denote the discs from the lowest one
up with the numbers i, 2, 3, . . . corresponding to the
stationary states 1, 2, 3, . . . and for the moment con¬
sider a fall from disc 6 to disc 5. We can now imagine
that the space between the two discs is in some way
tuned for a definite note. Thus we might place between
the discs a series of sheets of paper having such intervals
between them that the sphere in its fall strikes their
edges at equal intervals of time, e.g., 1/100 second.
The disturbance then set up will produce a sound with
the frequency 100 vibrations per second. If the distance
between the discs 5 and 4 double that between 6
and 5, the sphere in the fall from 5 to 4 will lose double
the energy lost in the descent from 6 to 5, and will
therefore emit a note of double frequency. The sheets
of paper in the space between 5 and 4 must then be
packed more tightly than between 6 and 5. And so
the space between any two discs may thus be said to
have its own particular classification or “ tuning.” In
analogy with this we might think of the space about a
hydrogen nucleus divided by the stationary states into
sections each with its own “ tuning.” But apart from

136 THE ATOM AND THE BOHR THEORY

the intrinsic peculiarity of such an arrangement and
the particular difficulties it will meet in trying to explain
the more complicated phenomena to be mentioned
later, the one fact that the electron in a transition from
one stationary state to another can jump over one or
more intervening stationary orbits, makes such a repre¬
sentation impossible. If the sphere in the given example
could fall from disc 6 to disc 4, it should during the
whole descent emit a note of higher pitch than in the
descent from 6 to 5. But this could not possibly take
place, if the space from 6 to 5, which must be traversed
en route to 4, is tuned for a lower note. The same
consideration applies to the hydrogen atom. Naturally
it is not impossible to continue the effort to illustrate
the matter in some concrete manner (one might, for
example, imagine separate channels each with its own
particular tuning between the same two discs). But in
all these attempts the situation must become more and
more complicated rather than more simple.

On the whole it is very difficult to understand how
a hydrogen atom, where the electron makes a transition
from orbit 6 to orbit 4, can during the entire transition
emit a radiation with a frequency different from that
when the electron goes from orbit 6 to orbit 5. Although
it seems as if the two electrons in making the transition
are at first under identical conditions, still, nevertheless,
the one which is going to orbit 4 emits from the first a
radiation different from that emitted by the one going
to orbit 5. Even from the very beginning the electron
seems to arrange its conduct according to the goal of
its motion and also according to future events. But
such a gift is wont to be the privilege of thinking beings
that can anticipate certain future occurrences. The

THEORY OF HYDROGEN SPECTRUM 137

inanimate objects of physics should observe causal
laws in a more direct manner, i.e., allow their conduct
to be determined by their previous states and the con¬
temporaneous influences on them.

There is a difficulty of a similar nature in the fact
that from the same stationary orbit the electron
sometimes starts for a single jump, another time for
a double jump, and so on. From certain considera¬
tions it is often possible to propound laws for the
probability of the different jumps, so that for a great
quantity of atoms it is possible to calculate the strengths
(intensities) of the corresponding spectral lines. But
we can no more give the reason why one given electron
at a given time determines to make a double jump
while another decides to make a single jump or not
to jump at all, than we can say why a certain radium
nucleus among many explodes at a given moment
(cf. p. 102). This similarity between the occurrence of
radiation processes on the Bohr theory and of the

radioactive processes has especially been emphasized

by Einstein.

It must, by no means, be said that the causal laws
do not hold for the atomic processes, but the hints
given here indicate how difficult it will be to reach
an understanding — in the usual sense — of these processes
and consequently of the processes of physics in general.
There is much that might indicate that, on the whole,
it is impossible to obtain a consistent picture of atomic
processes in space and time with the help of the motions
of the nuclei and the electrons and the variations in
the state of the ether, and with the application of such
fundamental conceptions of physics as mass, electric
charge and energy.

138 THE ATOM AND THE BOHR THEORY

Even if this were the case, it does not follow that a
comprehensive description in time and space of the
physical processes is impossible in principle ; but the
hope of attaining such a description must perhaps be
allied to the representation of “ physical individuals ”
or material particles of an even lower order of magnitude
than the smallest particles now known — electrons and
hydrogen nuclei — and to ideas of more fundamental
nature than those now known ; we are here outside
our present sphere of experience.

From all the above remarks it would be very easy
to get the impression that the Bohr theory, while it
gives us a glimpse into depths previously unsuspected,
at the same time leads us into a fog, where it is impos¬
sible to find the way. This is very far from being the
case. On the contrary, it has thrown new light on a host
of physical phenomena of different kinds so that they now
appear in a coherence previously unattainable. That
this light is not deceptive follows from the fact that the
theory, which has been gradually developed by Bohr and
many other investigators, has made it possible to predict
and to account for many phenomena with remarkable
accuracy and in complete agreement with experimental
observation. The fundamental concepts are, on the one
hand, the stationary states, where the usual laws of
mechanics can be applied (although only within certain
limits), and, on the other, the “ quantum rule ” for
transitions between the states. But at the very
beginning it, has been necessary in many respects to
grope in the dark, guided in part by the experimental
results and in part by various assumptions, often very
arbitrary.

For Bohr himself, a most important guide has been

THEORY OF HYDROGEN SPECTRUM 139

the so-called correspondence principle, which expresses
the previously mentioned connection with the classical
electrodynamics. It is difficult to explain in what it
consists, because it cannot be expressed in exact quanti¬
tative laws, and it is, on this account, also difficult to
apply. In Bohr’s hands it has been extraordinarily
fruitful in the most varied fields ; while other more
definite and more easily applicable rules of guidance have
indeed given important results in individual cases, they
have shown their limitations by failing in other cases.
We can here merely indicate what the correspondence
principle is.

As has been said (cf. p. 130), it has been found that in
the limiting region (sufficiently low frequencies) where
the Bohr theory and the classical electrodynamics are
merged in their outward features, a series of frequencies
plt v2, for monochromatic radiation, emitted by
different atoms in the single jumps, double jumps, etc.,
of the electrons, are equal to the frequencies m, 2m, 3m
. . . which, according to the laws of electrodynamics,
are contained in each of these atoms respectively as
fundamental and the first, second . . . overtones in the
motion of the electron. Farther away from this
region the two sets of frequencies are no longer equally
large, but it is easy to understand, from the foregoing, the
meaning of the statement that, for example, the radiation
of a triple jump with the frequency vz “ corresponds ”
to the second overtone 3 m in the revolution of the elec¬
tron. It is this correspondence which Bohr traces back
to the regions where there is even a great difference in
two successive orbits and where the frequency produced
by a transition between these orbits is very different
from the frequencies of revolution in the two orbits or

140 THE ATOM AND THE BOHR THEORY

their overtones. He expresses himself as follows : “ The
probability for the occurrence of single, double, triple
jumps, etc., is conditioned by the presence in the
motion of the atom of the different constituent
harmonic vibrations having the frequency of the
fundamental, first overtone, second overtone, etc.,
respectively.’ ’

In order to understand how this “ correspondence,”
apparently so indefinite, can be used to derive important
results, we shall give an illustration. Let us assume
that the mechanical theory for the revolution of an
electron in the hydrogen atom had led to the result that
the orbits of the electrons always had to be circles.
According to the laws of Electrodynamics, the motion of
the electron would in this case never give any overtones,
and, according to the correspondence principle, there
could not appear among the frequencies emitted by
hydrogen any which would correspond to the overtones,
i.e.} there would not be any double jumps, triple jumps,
etc., produced, but the only transitions would be those
between successive stationary orbits. The investigation
of the spectrum shows, however, that multiple jumps
occur as well as single jumps, and this fact may be taken
as evidence that the orbits in the hydrogen atom are not
usually circles. Let us next assume that, instead, we
had obtained the result that the orbits of the electrons
are always ellipses of a certain quite definite eccentricity,
corresponding to certain definite ratios in intensity
between the overtones and the fundamentals ; that, for
instance, the intensity of the classical radiation due to
the first overtone is in all states of motion always one-
half that due to the fundamental, the intensity due to the
second overtone always one-third that due to the funda-

THEORY OF HYDROGEN SPECTRUM 141

mental, etc. Then the radiation actually emitted should,
according to the correspondence principle, be such that
the intensities of the lines corresponding to the double
and triple jumps, which start from a given stationary
state, are respectively one-half and one-third of the in¬
tensity corresponding to a single jump from the same
state.

By these examples we can obtain an idea of how the
correspondence principle may in certain cases account
for various facts, as to what spectral lines cannot be
expected to appear at all, although they would be
given by a particular transition, and concerning the
distribution of intensities in those which really appear.
The illustration given above, however, has really nothing
much to do with actual problems, and objections may be
raised to the rough way in which the illustration has
been handled. The correspondence principle has its
particular province in more complicated electron motions
than those which appear in the unperturbed hydrogen
atom — motions which, unlike the simple elliptical motion,
are not composed of a series of harmonic oscillations
( co, 2a, 3& . . . ) but may be considered as compounded
of oscillations whose frequencies have other ratios.
The correspondence principle has, in such cases, given
rise to important discoveries and predictions which
agree completely with the observations.

We have dwelt thus long upon the difficult corre¬
spondence principle, because it is one of Bohr’s deepest
thoughts and chief guides. It has made possible a
more consistent presentation of the whole theory, and it
bids fair to remain the keystone of its future develop¬
ment. But from these general considerations we shall
now proceed to more special phases of the problem and

142 THE ATOM AND THE BOHR THEORY

examine one of the first great triumphs in which the
theory showed its ability to lead the way where pre¬
viously there had been no path.

The False Hydrogen Spectrum.

In 1897 the American astronomer, Pickering, dis¬
covered in the spectrum of a star, in addition to the usual
lines given by the Balmer series, a series of lines each
of which lay about midway between two lines of the
Balmer series ; the frequencies of these lines could be
represented by a formula which was very similar to the
Balmer formula ; it was necessary merely to substitute
n= 3|, 4J, 5 J, etc., in the formula on p. 57 instead of
n— 3, 4, 5, etc. It was later discovered that in many
stars there was a line corresponding to n" = 3/2 or w'=2
in the usual Balmer- Ritz formula (p. 59). It was con¬
sidered that these must be hydrogen lines, and that the
spectral formula for this element should properly be
written . I .

where n" and nr can assume integral values. This was
done since it was not to be believed that the spectral
properties of chemically different elements could be so
similar. This view was very much strengthened when
Fowler, in 1912, discovered the Pickering lines in the
light from a vacuum tube containing a mixture of
hydrogen and helium. It could not quite be understood,
however, why the new lines did not in general appear
in the hydrogen spectrum.

According to the Bohr theory for the hydrogen
spectrum it was impossible — except by giving up the
agreement (cf. p. 129) with electrodynamics in the region

THEORY OF HYDROGEN SPECTRUM 143

of high orbit numbers — to attribute to the hydrogen
atom the emission of lines corresponding to a formula
where the whole numbers were halved. The formula
given above might, however, also be written as

If the earlier calculations had been carried out a
little more generally, i.e., if instead of equating the
nuclear charge with i elementary electric quantum e ,
as in hydrogen, it had been equated with Ne where N
is an integer, then the frequency might have been
written as

This formula is evidently the same as that just given
when N equals 2. Now we know that helium has the
atomic number and nuclear charge 2 (cf. p. 90) ; a
normal neutral helium atom has two electrons and it is,
therefore, very different from a hydrogen atom. If,
however, a helium atom has lost one electron and there¬
fore has become a positive ion with one charge, it is a
system like the hydrogen atom with only one single
electron moving about the nucleus. It differs in its
“ outer ” characteristics from the hydrogen atom only
in having a nuclear charge twice as great, i.e. its spectral
formula must be given with N =2, or N2=4. The
formula for the supposed hydrogen lines would con¬
sequently fit the case of a helium atom which has
lost an electron. Bohr was aware of this, and he there¬
fore suggested that the lines in question were due, not
to hydrogen, but to helium.

At first all the authorities in the field of spectroscopy

144 THE ATOM AND THE BOHR THEORY

were against this view ; but most of the doubt was
dispelled when Evans showed that the lines could be
produced in a vacuum tube where there was only helium
with not a trace of hydrogen.

In a letter to Nature in September 1913, Fowler
objected to the Bohr theory on the ground that the
disputed line-formula did not exactly correspond to
the formula with 4K, but that there was a slight dis¬
agreement. Bohr’s answer was immediate. He called
attention to the fact that — since temporarily he had
sought only a first approximation — in his calculations
he had taken the mass of the nucleus to be infinite
in comparison to the mass of the electron, so that the
nucleus could be considered exactly at the focus of the
ellipse described by the electron. In reality, he said,
it must be assumed that nucleus and electron move
about their common centre of gravity, just as in the
motion in the solar system it must be assumed that not
the centre of the sun, but the centre of gravity of the
entire system remains fixed. This motion of the nucleus
leads to the introduction of a factor M/(M+m) in the
expression for the constant K given on p. 129, where
M is the mass of the nucleus and m that of the electron,
which in hydrogen is 1/1835 that of the nucleus. In
helium, M is four times as large as in hydrogen, so that
the given factor here has a slightly different value.
The difference in the values for K for the hydrogen
and for the helium spectrum which was found by Fowler,
is 0-04 per cent., which agrees exactly with the theoretical
difference.

Bohr thus turned Fowler’s objection into a strong
argument in favour of the theory.

THEORY OF HYDROGEN SPECTRUM

145

The Introduction of more than one Quantum Number.

During the first years after 1913, Bohr was practically
alone in working out his theory, at that time still
assailed by many, and in showing its application to
many problems. In 1916, however, the theorists in
other countries, led by the well-known Munich professor,
Sommerfeld, began to associate themselves with the
Bohr theory, and their investigations gave rise to much
essential progress. We shall here mention some of the
most important contributions.

In the theory for the hydrogen spectrum pro¬
pounded above, it was assumed that we had to do with
a single series of stationary orbits, each characterized
by its quantum number. But as shown by theoretical
investigations each of the stationary orbits must, when
more detail is asked for, also be indicated by an additional
quantum number.

This is closely connected with the fact that the
motion of the electron is not quite so simple as previously
assumed. We have assumed that the electron moves
about the nucleus just as a planet moves about the
sun (according to Kepler’s Laws), in an ellipse with
the sun at one focus, since the electron is influenced
by an attraction inversely proportional to the square
of the distance, just as the planets are attracted by the
sun according to Newton’s Law. We must, however,
remember that we are here concerned with the electric
attraction which at a given distance is determined, not
by mass, but by the electric charges in question. If
the latter remain unchanged, while the mass of the
electron varies, the motion will be changed, because the
same force has less effect upon a greater mass. Accord-

146 THE ATOM AND THE BOHR THEORY

in g to the Einstein principle of relativity, the mass of
the electrons, in accordance with ideas expounded long
ago by J. J. Thomson, will not be constant, but to a
certain extent depend upon the velocity, which will vary
from place to place, when the orbit is an ellipse. As a
result of this, the motion becomes a central motion of
more general nature than a Kepler ellipse. Since the

^ - ^

Fig. 26. — A compound electron motion produced by the
very rapid rotation of an elliptical orbit.

influence of the change of mass is very small, the orbit
can still be considered as an approximate Kepler
ellipse ; but the major axis will slowly rotate in the
plane of the orbit. In reality, the orbit will not there¬
fore be closed, but will have the character which is
shown in Fig. 26 ; this, however, corresponds to a
much more rapid rotation of the major axis than that
which actually takes place in the hydrogen atom,
where — even in case of the swiftest rotation — the
electron will revolve about 40,000 times round the

THEORY OF HYDROGEN SPECTRUM 147

nucleus at the same time as the major axis turns
round once.

If the electron moves in a fixed Kepler ellipse, the
energy content of the atom will be determined by the
major axis of the ellipse only. If these axes for the
stationary states with quantum numbers i, 2, 3 . . . are
respectively denoted by 2 av 2 a2, 2 a3 . . . the frequency,
for instance, in the transition from No. 3 to No. 2 —
since it is determined by the loss of energy — will be
the same whether the orbits are circles or ellipses. If,
on the other hand, the electron moves in an ellipse
which itself rotates slowly, the energy content, as can
be shown mathematically, will depend not only upon
the major axis of the ellipse, but also to a slight degree
upon its eccentricity, or, in other words, on its minor
axis. Then in the transition 3-2 we shall get different
energy losses and consequently different frequencies,
according as the ellipse is more or less elongated. If
it were the case that the eccentricity of the ellipses for a
certain quantum number could take arbitrary values,
then in the transition between two numbers we could
get frequencies which may take any value within a
certain small interval, i.e., a mass of hydrogen with its
great quantity of atoms would give diffuse spectral
lines, i.e. lines which are broadened over a small con¬
tinuous spectral interval. This is, however, not the
case ; but long before the appearance of the Bohr
theory it had been discovered that the hydrogen lines,
which we hitherto have considered as single, possess
what is called a fine structure. With a spectroscopic
apparatus of high resolving power each line can be
separated into two lying very close to each other.
This fine structure can now be explained by the fact

148 THE ATOM AND THE BOHR THEORY

that in a stationary state with quantum number 3
and major axis of the orbit 2a3, for instance, the eccen¬
tricity of the orbit has neither one single definite value
nor all possible values, but, on the contrary, it has
several discrete values of definite magnitude, to which
there correspond slightly different but definite values
of the energy content of the atom. It is now possible
to designate the series of stationary orbits, which have
the major axis 2 a3 with the principal quantum number 3,
with subscripts giving the auxiliary quantum number
for stationary orbits corresponding to the different
eccentricities, so that the series is known as 3X, 32, 33.
Instead of a single line corresponding to the transition
3-2, there are then obtained several spectral lines lying
closely together and corresponding to transitions such
as 33-22, 32-2!, etc. By theoretical considerations,
requiring considerable mathematical qualifications, but
of essentially the same formal nature as those Bohr
had originally applied to the determination of the
stationary orbits in hydrogen, Sommerfeld was led to
certain formal quantum rules which permit the fixing of
the stationary states of the hydrogen atom corresponding
to such a double set of quantum numbers. The results
he obtained as regards the fine structure of the hydrogen
lines agree with observation inside the limit of experi¬
mental error.

Although Sommerfeld’s methods have also been very
fruitful when applied to the spectra of other elements,
they were still of a purely formal and rather arbitrary
nature ; it is, therefore, of great importance that the
Leiden professor, Ehrenfest, and Bohr succeeded later
in handling the problem from a more fundamental point
of view, Bohr making use of the correspondence principle

THEORY OF HYDROGEN SPECTRUM 149

previously mentioned. It should be said here, by way
of suggestion, that Bohr used the fact that the motions
of the electrons are not simple periodic but “multiple
periodic.” We see this most simply if we think of the
revolution of the electrons in the elliptical orbit as repre-

Fig. 27. — 'The model of the hydrogen atom with stationary orbits
corresponding to principal quantum numbers and auxiliary
quantum numbers.

senting one period, and the rotation for the major axis
of the ellipse as representing a second period.

Fig. 27 shows a number of the possible stationary
orbits in the hydrogen atom according to Sommerf eld’s
theory ; for the sake of simplicity the orbits are drawn
as completely closed ellipses. If we examine, for
instance, the orbits with principal quantum number 4,
we have here three more or less elongated ellipses,
4i, 42, 43, and the circle 44 ; in all of them the major

150 THE ATOM AND THE BOHR THEORY

axis has the same length, and the length of the major
axis is to that of the minor axis as the principal quantum
number is to the auxiliary quantum number (for the
circle 4:4 = 1). On the whole, to a principal quantum
number n there correspond the auxiliary quantum
numbers 1, 2 . . . n, and the orbit for which the auxiliary
quantum number equals the principal quantum number
is a circle. We see that in the more complicated
hydrogen atom model there is possibility for a much
greater number of different transitions than in the
simple model (Fig. 25, p. 119). Some of the transitions
are indicated by arrows. Since the energy content of
the atom is almost the same for orbits with the same
principal quantum number and different auxiliary
quantum numbers, three transitions like 33-22, 32-2i
and 31-22 will give about the same frequency, and
therefore spectral lines which lie very close together.
In a transition like 44~43 the emitted energy quantum
hv , and also v, will be so extremely small that the
corresponding line will be too far out in the infra-red
for any possibility of observing it.

It must be pointed out that the above considerations
only hold if the hydrogen atoms, strictly speaking, are
undisturbed. Thus, very small external forces, which
may be due to the neighbourhood of other atoms, etc.,
will be sufficient to cause changes in the eccentricity of the
stationary orbits. In such a case the above definition
of the auxiliary quantum number becomes obviously
illusory, and the original character of the fine structure
disappears. This is in agreement with the experiments,
since the Sommerfeld fine structure can be found only
when the conditions in the discharge tube are especially
quiet and favourable.

THEORY OF HYDROGEN SPECTRUM 151

Influence of Magnetic and Electric Fields on the Hydrogen
Lines.

As previously mentioned (p. 76), the spectral lines
are split into three components when the atoms emitting
lights are exposed to magnetic forces. The agreement
found here between observation and the Lorentz
electron theory was considered as strong evidence of

Fig. 28. — The splitting of three hydrogen lines under the influence

of a strong electric field.

the correctness of the latter. According to the Bohr
theory, the picture upon which this explanation rested
must be abandoned entirely ; but fortunately it has
been shown that the Bohr theory leads to the same
results ; and, moreover, Bohr, with the assistance of the
correspondence principle, has been able to set forth the
more fundamental reason for this agreement.

The German scientist, Stark, showed, in 1912, that
hydrogen lines are also split by electric fields of force.
In Fig. 28 it is shown how very complicated this pheno-

152 THE ATOM AND THE BOHR THEORY

menon is ; here the classical electron theory could not
at all explain what happened. This phenomenon could
also be accounted for by the extended Bohr theory
(with the introduction of more than one quantum
number), as it was shown independently by Epstein
and by Schwarzschild in 1916 ; further, the correspon¬
dence principle has again shown its superiority, since it
makes possible an approximate determination of the
different intensities of the different lines. A calculation
carried out by H. A. Kramers has shown that the
theory gives a remarkably good agreement with the
experiments.

Not until we think of the extraordinary accuracy of
the measurements which are obtained by spectrum
analysis, can we thoroughly appreciate the importance
of the quantitative agreement between theory and
observation in the hydrogen spectrum that has just
been mentioned. Moreover, we must remember how
completely helpless we previously were in the strange
puzzles offered even by the simplest of all spectra, that
of hydrogen.

CHAPTER VI

VARIOUS APPLICATIONS OF THE BOHR

THEORY

Introduction.

We have dwelt at length upon the theory of the
hydrogen spectrum because it was particularly in this
relatively simple spectrum that the Bohr theory first
showed its fertility. Moreover, by studying the case
of the hydrogen atom with its one electron, it is easier
to gain insight into the fundamental ideas of the Bohr
theory and its revolutionary character. Naturally, the
theory is limited neither to the hydrogen atom nor to
spectral phenomena, but has a much more general
application. As has already been said, it takes, as its
problem, the explanation of every one of the physical
and chemical properties of all the elements, with the
exception of those properties known to be nuclear
(cf. p. 94). This very comprehensive problem can
naturally, even in its main outlines, be solved but
gradually and by the co-operation of many scientists,
and it is quite impossible to go very deeply into the
great work which has already been accomplished, and
into the difficulties which Bohr and the others working
on the problem have overcome. We must be content
with showing some especially significant features.

153

154 THE ATOM AND THE BOHR THEORY

Different Emission Spectra.

While the neutral hydrogen atom consists simply
of a positive nucleus and one electron revolving about
the nucleus, the other elements, in the neutral state,
have from two up to 92 electrons in the system of
electrons revolving around the nucleus. Even 2 elec¬
trons, as in the helium atom, make the situation far
more complicated, since we have in this case a system
of 3 bodies which mutually attract or repel each other.
We are thus confronted with what, in astronomy, is
known as the three-body problem, a problem considered
with respect by all mathematicians on account of its
difficulties. In astronomy, the difficulties are restricted
very much when the mass of one body is many times
greater than that of the others, as in the case of the
mass of the sun in relation to that of the other planets.
Here, by comparatively simple methods, it is possible
to calculate the motions inside a finite time-interval
with a high degree of approximation even when there
are not two but many planets involved.

We might now be tempted to believe that in the
atom we had to deal with comparatively simple systems
— solar systems on small scale — since the mass of the
nucelus is many times greater than that of the electrons.
But even if the suggested comparison illustrates the
position of the nucleus as the central body which holds
the electrons together by its power of attraction, the
comparison in other respects is misleading. While
the orbits of the planets in the solar system may be at
any distance whatsoever from the sun, and the motions
of the planets are everywhere governed by the laws
of mechanics, the atomic processes, according to the

APPLICATIONS OF THE BOHR THEORY 155

Bohr theory, are characterized by certain stationary
states, and only in these can the laws of mechanics
possibly be applied. But in addition, the forces between
nucleus and electrons are determined not at all by the
masses, but rather by the electric charges. In the
helium atom the nuclear charge is only double that of
an electron, and the attraction of the nucleus for an
electron will therefore be only twice as large as the
repulsions between two electrons at the same distance
apart. This repulsion under these circumstances will,
therefore, also have great influence on the ensuing
motion. In elements with higher atomic numbers the
nuclear charge has greater predominance over the
electron charges ; but, on the other hand, there are then
more electrons. The situation is in each case more
complicated than in the hydrogen atom.

Nevertheless, the line spectra of the elements of
higher atomic number show how the lines, as in the
hydrogen spectrum, are arranged in series although in
a more complicated manner (cf. p. 59) ; in any case
in many instances there is great similarity between the
radiation from the hydrogen atom and that from the
more complicated atoms. Thus in the line spectra of
many elements, just as in that of hydrogen, the frequency
v of every line can be expressed as a difference between
two terms , involving certain integers which can pass
through a series of values. From the combinations of
terms, two at a time, the values of v corresponding to
the different spectral lines can be derived. This so-
called combination principle enunciated by the Swiss
physicist, Ritz, can evidently be directly interpreted on
the basis of Bohr’s postulates, since the different com¬
binations may be assumed to correspond to definite

156 THE ATOM AND THE BOHR THEORY

atomic processes, in which there is a transition between
two stationary states, each of which corresponds to a
spectral term.

Moreover, the terms (cf. p. 59) may often be
approximately given by the Rydberg formula

(w + a)2

where K has about the same value as in hydrogen, and
a can take on a series of values otv cc2 . . . uk, while n
takes on integer values. Since we thus determine the
different lines by assigning values to the two integers
n and k in each term, we have in this respect something
like the fine structure in the hydrogen spectrum, where
the stationary states are determined by a principal
quantum number and an auxiliary quantum number.
The spectra of which we are speaking here, and for
which the terms have the form given above, are often
called arc spectra, because they- are emitted particularly
in the light from the electric arc or from the vacuum
tube. We must expect that the similarity which exists
in the law for the distribution of spectral lines will
correspond to a similarity in the atomic processes of
hydrogen and the other elements.

The hydrogen atom emits radiation corresponding
to the different spectral lines when an electron from an
outer stationary orbit jumps, with a spring of varying
size, to an orbit with lower number, and at last finds rest
in the innermost orbit in a normal state, where the energy
of the atom is as small as possible. Similarly, we must
assume that the electrons in other atoms, during processes
of radiation, may proceed in towards the nucleus until
they are collected as tightly as possible about the

APPLICATIONS OF THE BOHR THEORY 157

nucleus, corresponding to the normal state of the atom,
where its energy content is as small as possible :
“ capture ” of electrons by the nucleus. The region in
space which, in the normal state, includes the entire
electron system, must be assumed to be of the same order
of magnitude as the dimensions of the atom and molecule
which are derived from the kinetic theory of gases.
This normal state may be called a “ quiescent ” state,
since the atom cannot emit radiation until it has been
excited by the introduction of energy from without.
This excitation process consists of freeing one (or more)
electrons, in some way or other, from the normal state
and either removing it out to a stationary orbit farther
away from the nucleus or ejecting it completely from
the atom. Not all electrons can be equally easily
removed from the quiescent state. Those moving in
small orbits near the nucleus will be tighter bound
than those moving in larger orbits farther from the
nucleus. The arc spectrum is now caused by driving
one of the most loosely bound electrons out into an
orbit farther from the nucleus or removing it com¬
pletely from the atom. In the latter case the rest of
the atom, which with the loss of the negative electron
becomes a positive ion, easily binds another electron,
which, with the emission of radiation, corresponding to
lines of the series spectrum, can approach closer to
the nucleus.

Let us now assume, first, that this radiating electron
moves at so great a distance from the nucleus and the
other electrons that the entire inner system can be
considered as concentrated in one point ; then the
situation is quite as if we had to deal with a hydrogen
atom. If the atomic number is as high as 29 (copper),

158 THE ATOM AND THE BOHR THEORY

for instance, the nuclear charge will consist of twenty-nine
elementary quanta of positive electricity ; but since
there are twenty-eight electrons in the inner system,
the resultant effect is that of only one elementary
quantum of positive electricity, as in the case of a
hydrogen nucleus. The spectral lines which are emitted
in the jumps between the more distant paths will be
practically the same as hydrogen lines. But, since in
the jumps between these distant orbits, very small
energy quanta will be emitted, the frequencies are very
small, the wave-lengths very great, i.e., the lines in
question lie far out in the infra-red.

When the electron has come in so close to the nucleus
that the distances in the inner system cannot be assumed
to be small in comparison to the distance of the outer
electron from the nucleus, the situation is changed.
The force with which the nucleus and the inner electrons
together will work upon the outer electron will be appreci¬
ably different from the inverse square law of attraction
of a point charge. The consequence of this difference
is that the major axis in the ellipse of the electron
rotates slowly in the plane of the orbit as described in
case of the theory of the fine structure of the hydrogen
lines (cf. p. 146), and even if the cause is different the
result is the same ; the orbit of the outer electron
in the stationary states will be characterized by a
quantum number n and an auxiliary quantum number k.
If the electron comes still closer to the nucleus, its
motion is even more complicated. When the electron
in its revolution is nearest the nucleus it will be able to
dive into the region of the inner electrons, and we can
get motions like those shown in Fig. 29 for one of the
eleven sodium electrons. The inner dotted circle is

APPLICATIONS OF THE BOHR THEORY 159

the boundary of the inner system which is given by
the nucleus and the ten electrons remaining in the
“ quiescent ” state — little disturbed by the restless

Fig. 29. — Different stationary orbits which the outermost (nth)
electron of sodium may describe.

No. 11. In the figure we can see greater or smaller
parts of No. n’s different stationary orbits with
principal quantum numbers 3 and 4. We shall not
account further for the different orbits and the spectral
lines produced by the transitions between orbits, but

160 THE ATOM AND THE BOHR THEORY

shall merely remark that the yellow sodium line, which
corresponds to the Fraunhofer D-line (cf. p. 49) >
produced by the transition 32-3i, between two orbits
with the same principal quantum number. The sketch
shows to a certain degree how fully many details of
the atomic processes can already be explained. The
theory can even give a natural explanation of why the

D-line is double.

We have restricted ourselves to the case where only
one electron is removed from the normal state of the
neutral atom. It may, however, happen that two elec¬
trons are ejected from the atom so that it becomes a
positive ion with two charges. When an electron from
the outside is approaching this doubly charged ion it
will, at a distance, be acted upon as if the ion were a
helium nucleus with two positive charges. The situa¬
tion, in other words, will be as in the case of the false
hydrogen spectrum (cf. p. I42)> where the constant K
in the formula for the hydrogen spectrum is replaced by
another which is very close to 4^* But if the atom is
not one of helium, but one with a higher atomic number,
the stationary orbits of the outer electron which approach
closely to the nucleus will not coincide exactly with those
in the ionized helium atom, corresponding to the fact
that the terms in the formula for the spectrum, instead
of the simple form 4K/#2, have the more complicated
form 4K /(w + a)2. Spectra of this nature are often
called spark spectra, since they appear especially strong in
electric sparks ; they appear also in light from vacuum
tubes, when an interruptor is placed in the circuit,
making the discharge intermittent and more intense.

An atom with several electrons can, however, be
much more violently excited from its quiescent state

APPLICATIONS OF THE BOHR THEORY 161

when an electron in the inner region of the atom is
ejected by a swiftly moving electron (a cathode ray
particle or a /3-particle from radium) which travels
through the atom. Such an invasion produces a serious
disturbance in the stability of the electron system ; a
reconstruction follows, in which one of the outer, more
loosely bound electrons takes the vacant position. In
the transitions, in which these outer electrons come in,
rather large energy quanta are emitted. The emitted
radiation has therefore a very high frequency ; mono¬
chromatic X-rays are thus emitted. Since these have
their origin in processes far within the atom, it can be
understood that the different elements have different
characteristic X-ray spectra, which can give very
valuable information about the structure of the electron
system (cf. p. 91).

Between these X-ray spectra and the series spectra
previously mentioned there lie, as connecting links,
those spectra which are produced when electrons are
ejected from a group in the atom which does not belong
to the innermost group, but does not, on the other hand,
belong in the outermost group in the normal atom. We
have very little experimental knowledge about such
spectra, because the spectral lines involved have wave¬
lengths lying between about 1*5 and 100 fjbfju. Rays
with these wave-lengths are absorbed very easily by all
possible substances ; they have very little effect on
photographic plates, where they are absorbed by the
gelatine coating before they have an opportunity to
influence the molecules susceptible to light. But there
can be scarcely any doubt that, in the course of a few
years, experimental technique will have reached such
efficiency that this domain of the spectrum, so important

162 THE ATOM AND THE BOHR THEORY

for the atomic theory, will also become accessible to
experiment. In individual cases, wave-lengths as small
as 20 [Jj[l have already been obtained by Millikan.

Of entirely different character from these spectra
are the band spectra. They are in general produced by
electric discharges through gases which are not very
highly attenuated (cf. p. 55) ; they are not due to purely
atomic processes, but can be designated as molecular
spectra. Their special character is due to motions in the
molecule, not only motions of the electrons, but also oscil¬
lations and rotations of the nuclei about each other.
We shall not go into these problems here ; in what
follows we shall investigate a certain type of band
spectra somewhat more closely in connection with the
absorption of radiation.

While the band spectra with a spectroscope of high
resolving power can be more or less completely resolved
into lines, this is not the case with the continuous spectra.
They are emitted not only by glowing solids (cf. p. 54),
but also by many gaseous substances. When such
gases are exposed to electric discharges they emit, in
addition to the line spectra and band spectra, continuous
spectra which in certain parts of the spectrum furnish a
background for bright lines which come out more strongly.
It might seem impossible to correlate these with the
Bohr theory ; but in reality a spectrum does not always
have to consist of sharp lines. This can at once be seen
from the correspondence principle. If the motions in the
stationary states are of such nature that they can be
resolved into a number of discrete harmonic oscillations
each with its own period (for instance the orbit of an
electron in a rotating ellipse ; cf. p. 149), then, according
to the correspondence principle, in the transition

APPLICATIONS OF THE BOHR THEORY 163

between two such stationary states there are produced
sharp spectral lines “ corresponding ” to these harmonic
components. But not all motions of atomic systems
can be thus resolved into a number of definite harmonic
oscillations. When this cannot be done, the stationary
states cannot be expected to be such that transitions
between them produce radiation which can be resolved
into sharp lines.

A simple example, where it is easily intelligible that
the Bohr theory will not lead to sharp lines, is obtained
in a simple consideration of the hydrogen atom. Let
us examine the lines belonging to the Balmer series
which are produced when an electron passes to the No. 2
orbit from an orbit with higher orbit number, which is
farther from the nucleus. As has been said, we obtain
here an upper limit for the frequency corresponding to a
value of the outer orbit number which is infinite ; this
means, in reality, that the electron in one jump comes
in from a distance so great that the attraction of the
nucleus is infinitely small. The energy released by such
a jump is the same as the ionizing energy A2 which is
required to eject the electron from the orbit No. 2 and
drive it from the atom. It is here assumed, however,
that the electron out in the distance was practically at
rest. If the captured electron has a certain initial
velocity outside, it will have a corresponding kinetic
energy A. When in one jump this electron comes from
the outside into orbit No. 2, the energy lost by the
electron and emitted in the form of radiation will be the
sum of the ionizing energy A2 and the original kinetic
energy A. The frequency v will then become greater than
that corresponding to A2 ; and since the velocity of the
electron before it is captured is not restricted to certain

164 THE ATOM AND THE BOHR THEORY

definite values, neither is the value of v. The radiation
from a great quantity of hydrogen atoms which are
binding electrons in this way will, in the spectrum, not
be concentrated in certain lines, but will be distributed
over a region in the ultra-violet which lies outside of the
limit calculated from the Balmer formula ; still in a
certain sense this continuous spectrum is correlated with
the Balmer series. In the spectra from certain stars
there has actually been discovered a continuous spectrum,
which lies beyond the limits of the Balmer series and
may be said to continue it.

Also the X-rays, which are generally used in medicine,
have varying frequencies ; this is caused by the fact that
some of the electrons which, in an X-ray tube, strike the
atoms of the anticathode and travel far into it at a high
speed, lose a part or all of their velocity without ejecting
inner electrons. The lost kinetic energy then appears
directly as radiation. These remarks ought to be sufficient
to show that the radiation, for instance, from a glowing
body, where the interplay of atoms and molecules is very
complicated, can give a continuous spectrum.

Electron Collisions.

The excitation of an atom in the normal state (cf. p.
157), by which one of its electrons is removed to an outer
stationary orbit, may be caused by a foreign electron
which strikes the atom. A study of collisions between
atoms and free electrons is therefore of the greatest
importance when investigating more closely the condi¬
tions by which series spectra are produced.

These investigations can be carried out by giving
free electrons definite velocities by letting them pass
through an electric field, where the “ difference of

APPLICATIONS OF THE BOIIR THEORY 105

potential ” is known in the path traversed by the

electrons. When an electron moves through a region

with a difference of potential of one volt (the usual

technical unit), the kinetic energy of the electron will

be increased by a definite amount (of i-6xicr12 erg). !

If its initial velocity is zero, its passage through this 1

field will make the velocity 600 km. per second ; if the

potential difference were 4 volts, 9 volts, etc., the velocity

obtained by the electron would be 2, 3, etc., times larger.

For the sake of brevity we shall say that the kinetic
energy of an electron is, for instance, 15 volts, when we
mean that the kinetic energy is as great as would be
given by a difference of potential of 15 volts.

In 1913 the German physicist Franck began a series
of experiments by methods which made it possible to
regulate accurately the velocity of the electrons, and to
determine the kinetic energy before and after collisions
with atoms. He first applied the methods to mercury
vapour, where the conditions are particularly simple,
since the mercury molecules consist of only one atom.

Franck bombarded mercury vapour with electrons all
of which had the same velocity. He then showed that
if the kinetic energy of the electrons was less than 4-9
volts the collisions with the atoms were completely
“ elastic,” i.e.} the direction of the electron could be
changed by the collision, but not its velocity. If, how¬
ever, the velocity of the impinging electrons was in¬
creased so much that it was somewhat larger than 4-9
volts, there was an abrupt change in the situation,
since many of the collisions became completely inelastic,
i.e., the colliding electron lost its entire velocity and gave
up its entire kinetic energy to the atom. If the initial
velocity was even greater, so that the kinetic energy of

166 THE ATOM AND THE BOHR THEORY

the colliding electron was 6 volts, for instance, then when
the collision took place there would always be lost a
kinetic energy of 4*9 volts, since the electrons would
either preserve their kinetic energy intact or have it
reduced to i-i volt (cf. Fig. 30).

This remarkable phenomenon can be understood

6 Volt.

Fig. 30. — Schematic drawing of Franck's experiment with electron
collisions. G is a glowing metal wire which emits electrons.
If between G and the wire net T there is a difference of
potential of 6 volts, the electrons will pass through the holes
of the net with great velocity out into the space R, where
there is mercury vapour, a represents a free electron F and
a mercury atom Hg before the collision, while b represents
them after the collision ; with the collision F loses a kinetic
energy corresponding to 4-9 volts ; at the same time a bound
electron B in the atom goes over to a larger stationary orbit.

from the Bohr theory if we assume that to send the most
loosely bound electron in the mercury atom out to the
nearest outer stationary orbit there is required an
energy of 4-9 volts, since in that case, according to the
first postulate, an energy of less than this magnitude
cannot be absorbed by the atom. The use of the word
“ understanding ” must here be qualified ; if the forces
which influence the free electron as it comes into the

APPLICATIONS OF THE BOHR THEORY 167

electron system of the mercury atom are no other than
the usual repulsion from the electrons and the attraction
from the nucleus, the conduct of the colliding electron can
in no way be explained by the laws of mechanics. But
what happens is in agreement with the characteristic
stability of the stationary states, and Bohr had pro¬
phesied how it would happen. Curiously enough Franck
believed in the beginning that his experiment disagreed
with the Bohr theory because he made the mistake of
supposing that what happened was merely ionization,
i.e., complete disruption of a bound electron from a
mercury atom.

Franck’s experiments showed, moreover, that mercury
vapour, as soon as the inelastic collisions appeared, began
to emit ultra-violet light of a definite wave-length,
namely, 253*7 ft ft- The product of the frequency v of
this light and Planck’s constant h agrees exactly with
the energy quantum possessed by an electron which
has passed a potential difference of 4-9 volts ; but this
also agrees with what might be expected, according to
the Bohr theory, from the radiation the removed electron
would emit upon returning to the normal state. The
energy which is respectively absorbed and emitted in
the two transitions must be indeed hv.

Since an electron can not only be driven out to the
next stationary orbit, but also to an even more distant
one (or entirely ejected) and thence can come in again
in one or more jumps, it is evident that a far more
complicated situation may arise. The Franck experi¬
ment, which now has been extended to many othei
elements, clearly gives extraordinarily valuable informa¬
tion in such cases. In mercury it has been found that
the energy a free electron must have in order to eject an

168 THE ATOM AND THE BOHR THEORY

electron from an atom and turn the atom into a positive
ion, corresponds to a difference of potential of io*8 volts,
a value which Bohr had predicted. At the same time
that Franck’s experiments, in this respect and in others,
have strengthened the Bohr theory in the most satis¬
factory way, they have also advanced its development
very much. Indeed it may be said that they have been
of the greatest help in atomic research. Even if the
spectroscope has greater importance, the investigations
on electron collisions make the realities in the Bohr
theory accessible to study in a more direct and palpable
manner.

Fig. 31. — Stratification of light in a vacuum tube.

The peculiarities in the electron collisions appear
most clearly in an old and well-known phenomenon of
light, namely, the stratification of the light in a vacuum
tube (Fig. 31). This stratification, which previously
seemed so incomprehensible, agrees exactly with the
feature so fundamental in the atomic theory that a free
electron cannot give energy under a certain quantum to
an atom. We can imagine that, in the non-illuminated
central space between the bright strata, the electrons
each time under the influence of the outer electric field
obtain the amount of kinetic energy which must serve to
excite the atoms of the attenuated vapour.

As has been said (p. 161) , electron collisions may cause
the emission of characteristic X-rays ; but to produce

v ■

APPLICATIONS OF THE BOHR THEORY 169

them very great energy is required. Therefore the elec¬
trons which are to produce this effect must have an
opportunity to pass freely through a certain region under
the influence of a proportionately strong electric field
(with potential of from 1000 to 100,000 volts and more).
The electrons find such a field in a highly exhausted
X-ray tube, where the electrons under strong potential
are driven from the cathode against the anti-cathode,
into which they penetrate deeply.

Absorption.

In the experiments previously described it was the
electron collisions which furnished the energy required
to excite the atoms, i.e., to carry them from the normal
state over into a stationary state with greater energy.
This “ excitation energy ” may, however, also be
furnished to the atoms in the form of radiation energy ;
we shall now examine this case more closely.

Let us assume that to transfer an atom from the
normal state to another stationary state, or, in other
words, to transfer one of the electrons to an outer sta¬
tionary orbit, a certain quantity of energy E is demanded ;
then the radiation emitted by the atom when it returns
to the normal state will have a frequency v depending
upon the relation E —hv or j/=E/A, where h, as usual,
is the Planck constant. But just as the atom in the
transition from the stationary state to the normal state
can emit radiation only with the definite frequency v, then
the opposite transition can only be performed by absorp¬
tion of radiation with the same frequency ; when this
happens the absorbed radiation energy has exactly the
value E —hv.

This reciprocity, which may be considered as a

170 THE ATOM AND THE BOHR THEORY

direct consequence of the Bohr postulates, agrees with
what has been said (cf. p. 50) about the correspon¬
dence between the lines in the line spectrum of an
element and the dark absorption lines of that element —
e.g., the Fraunhofer lines in the solar spectrum. Let
us examine, as an example, the yellow sodium line,
the D-line. Light with the corresponding frequency,
526 X io12 vibrations per second, is emitted by a sodium
atom, when the loosest bound electron goes over from a
stationary orbit with quantum numbers 32 to the orbit 3X,
which belongs to the normal state of the sodium atom.
The transition in the opposite direction, 3X to 32, can
take place under absorption of radiation only when in
the light from some other source of light, which passes
the sodium atoms, there are found rays with the fre¬
quency 526 X io12. Even if there is present radiation
energy with some other frequency, the sodium atoms
take no notice of this energy ; they ‘absorb only rays
with the frequency stated, and every time an atom
absorbs energy from a ray the energy taken is always
an energy quantum of the magnitude hv, i.e. about
6-54 x io-27 x 526 x io12 =3 44 x io-12 ergs (1 erg is the unit
of energy used in the determination of h). When there
are present a large number of sodium atoms (as, for
instance, in the previously mentioned common salt
flame), the transition 3X to 32 can take place in some
atoms, the transition 32 to 3X in others ; therefore, at
the same time there can be absorption and radiation
of the light in question. Whether absorption or
radiation at any given time has the upper hand depends
upon various conditions (temperature, etc.).

For the sake of simplicity we have here tacitly
understood that there can be but one definite transition

APPLICATIONS OF THE BOHR THEORY 171

(from the normal state) corresponding to the assumption
that the sodium spectrum had no other lines than the
D-line. In reality this is not the case, and there can
equally occur absorption of rays with larger frequencies
belonging to other spectral lines in the sodium atom and
corresponding to other possible transitions between
stationary states in the sodium atom. If the tem¬
perature of the sodium vapour is sufficiently low, in
which case almost all the atoms are in the normal state,
it is evident that in the absorption only those lines will
appear which correspond to transitions from the normal
state, and which therefore form only a part of all the
lines of the sodium spectrum. We thus obtain an ex¬
planation of the previously enigmatical circumstance that
not all spectral lines which can appear in emission will
be found in absorption. At the same time we get, in
absorption experiments, valuable information about the
structure of the atom beyond what the observations in
the emission spectra are able to give.

Interesting phenomena may arise owing to the fact
that the jumps between the stationary states of the atom
sometimes, as we know, take place in single jumps,
sometimes in double or multiple jumps, so that the inter¬
mediate stationary states are jumped over. There is
then evidently a possibility that absorption can take
place, for instance, with a double jump of an electron,
which may later return to the original stationary orbit
in two single jumps. The absorbed radiation energy
will then appear in emission with two frequencies which
are entirely different from the frequency of the absorbed
rays (this latter in this case will be the sum of the other
two). When an element is illuminated with a certain
kind of rays, it can, in other words, emit in return rays

172 THE ATOM AND THE BOHR THEORY

of a different nature. Such changes of frequencies have
also been observed in experiment ; they contain, in
principle, an explanation of the characteristic phenom¬
enon called fluorescence.

We shall not go further into this problem, but dwell
for a time on the characteristic phenomenon of ab¬
sorption which is known as the photo-electric effect. In
this phenomenon (cf. p. 116) a metal plate, by illumina¬
tion with ultra-violet light, is made to send out electrons
with velocities the maximum value of which is in¬
dependent of the strength of the illumination, but
depends only on the frequency of the rays. What
happens is that some of the electrons in the metal which
otherwise have, as their function, the conduction of
the electric current, by absorbing radiation energy, free
themselves from the metal and leave it with a certain
velocity. The reason why the rays for most metals
must be ultra-violet (i.e. have a high frequency and
consequently correspond to a proportionately large
energy quantum) depends upon the fact that the energy
quantum absorbed by the electrons must be large
enough to carry out the work of freeing the electrons.
But as long as the frequency of the rays (and therefore
their energy quantum) is no less than what is needed for
the freeing process, it does not need to have certain fixed
values. If the energy quantum hv which the rays can
give off is greater than is required to free the electrons,
the surplus becomes kinetic energy in the electrons,
which thus acquire a velocity which is the greater the
greater the frequency v, and which coincides with the
maximum velocity observed in the experiments. What
happens here is evidently something which can be
considered as the reverse of the process which leads

APPLICATIONS OF THE BOHR THEORY 173

to the production of the continuous hydrogen spectrum
(described on p. 163). In the latter case, electrons with
different velocities are bound by the hydrogen atoms,
which thus emit rays with frequencies increasing with
increasing velocity, while, vice versa, in the photo¬
electric effect rays with different frequencies free the
electrons and give them velocities increasing with
increasing frequencies.

It must be acknowledged that there is something
very curious in this effect. If the electromagnetic
waves, as has been assumed, are distributed evenly over
the field of radiation, it is not easy to understand why
they give energy to some atoms and not to others, and
why the selected ones always — with a given frequency
— acquire a definite energy quantum, independent of the
intensity of the radiation. For small intensities of the
incident radiation, the atom, in order to acquire the
proper quantum, must absorb energy from a greater
part of the field of radiation (or for a longer time) than
for large intensities. When the atoms acquire energy
in electron collisions, the situation is apparently easier
to understand, since in this case the colliding electrons
give their kinetic energy to definite atoms, namely,
those which they strike.

Einstein, in 1905, when there was not yet any talk
of the nuclear atom or the Bohr theory, enunciated his
theory of light quanta, according to which the energy
of radiation is not only emitted and absorbed by the
atoms in certain quanta, with magnitudes determined
by the frequency v, but is also present in the field of
radiation in such quanta. When an atom emits an
energy quantum hv, this energy will not spread out in
waves on all sides, but will travel in a definite direction

174 THE ATOM AND THE BOHR THEORY

— like a little lump of energy, we might say. These light
quanta, as they are called, can, like the electrons, hit
certain atoms.

But even if in this theory the difficulties mentioned
are, apparently, overcome, far greater difficulties are
introduced ; indeed it may be said that the whole wave
theory becomes shrouded in darkness. The very
number v which characterizes the different kinds of rays
loses its significance as a frequency and the phenomena of
interference — reflection, dispersion, diffraction, and so
on — which are so fundamental in the wave theory of
the propagation of light, and on which, for instance, the
mechanism of the human eye is based, receive no explana¬
tion in the theory of light quanta.

For instance, in order to understand that grating
spectra can be produced at all, we must think of a co¬
operation of the light from all the rulings (cf. Fig. io, p. 47),
and this co-operation cannot arise if all the slits at a
given moment do not receive light emitted from the same
atom. In a bundle of rays which comes in at right angles
to a grating, we must, in order to explain the inter¬
ference, assume that the state of oscillation at a given
moment is the same in all slits, that, for instance, there
are wave crests in all at the same time, if we borrow a
picture from the representation of water waves. Only
in this case there can behind the grating at certain
fixed places — for which the difference in the wave¬
length of the distances from successive slits is a whole
number of wave-lengths — steadily come wave crests
from all the slits at one moment and wave troughs from
all at another moment (the classical explanation of the
“ mechanism ” of a grating). If we imagine, however,
that some slits are hit by light quanta from one atom

APPLICATIONS OF THE BOHR THEORY 175

and others from a second atom, it is pure chance if there
are wave crests simultaneously in all slits, because the
different atoms in a source of light emit light at different
times, depending purely on chance. An understanding
of the observed effect of a grating on light seems then out
of question.

The theory of light quanta may thus be compared
with medicine which will cause the disease to vanish
but kill the patient. When Einstein, who has made so
many essential contributions in the field of the quantum
theory, advocated these remarkable representations
about the propagation of radiation energy he was natur¬
ally not blind to the great difficulties just indicated.
His apprehension of the mysterious light in which the
phenomena of interference appear on his theory is shown
in the fact that in his considerations he introduces some¬
thing which he calls a “ ghost ” field of radiation to help
to account for the observed facts. But he has evidently
wished to follow the paradoxical in the phenomena of
radiation out to the end in the hope of making some
advance in our knowledge.

This matter is introduced here because the Einstein
light quanta have played an important part in discussions
about the quantum theory, and some readers may have
heard about them without being clear as to the real
standing of the theory of light quanta. The fact must
be emphasized that this theory in no way has sprung
from the Bohr theory, to say nothing of its being a
necessary consequence of it.

In the Bohr theory, absorption and radiation must be
said to be completely reciprocal processes, i.e. processes
of essentially the same nature, but proceeding in oppo¬
site directions. In itself it cannot be said to be more

176 THE ATOM AND THE BOHR THEORY

incomprehensible that an atom absorbs energy from a
field of radiation in agreement with the Bohr postulates
than that it emits energy into the field ; but in both
cases we naturally encounter the great difficulties men¬
tioned in Chap. V.

We have hitherto restricted ourselves to the purely
atomic processes. But just as in the emission of radia¬
tion we meet spectra which owe their characteristics to
molecular processes (band spectra, cf. p. 162), we have
also absorption spectra with characteristics depending
essentially upon motions of the atomic nuclei in the
molecules. A particularly interesting and instructive
example of this nature is met with in the infra-red
region of the spectrum in certain broad absorption lines
or absorption bands, which are due to gases having
molecules containing several atoms. In hydrogen
chloride, for instance, there is found, in the region of the
spectrum which corresponds to a wave-length of about
3-5 (A, such an absorption band, which by more accurate
investigation has been shown to consist of a great number
of absorption lines.

The explanation of this collection of lines must be
sought in the motions which the hydrogen nucleus and
the chlorine nucleus perform, as they in part vibrate with
respect to each other and in part rotate about their com¬
mon centre of gravity. Just as in the case of the motions
of the electrons in the atom, there are also certain
stationary states for the nuclear motions. When the
molecule absorbs radiation energy it will go from one of
these states to another, where the energy content is
greater. This absorption of energy proceeds according
to the quantum rule, i.e., the product of the Planck
constant h and the frequency v for the absorbed radiation

APPLICATIONS OF THE BOHR THEORY 177

must be equal to the difference in energy between the
two stationary states ; only those rays which have
frequencies fulfilling this condition are absorbed.

In hydrogen chloride, at standard temperature, the
molecules will be in different stationary states of rotation
(cf. the remarks on p. 27), corresponding to different
definite values of the rotation frequency, while the
nuclei, on the other hand, must be assumed to be at
rest with reference to each other, i.e.} they preserve their

Fig. 32. — Schematic representation of possible motions in
a molecule of hydrogen chloride. 0 is the centre
of gravity of the molecule. The black circles give
the states of equilibrium of the nuclei, the circles s
their outer positions in oscillating, and the circles r
positions during the rotation of the nuclei.

- «

mutual distance. In Fig. 32, H and Cl indicate the
circles which the two nuclei will describe about the centre
of gravity ; here, however, it must be remarked that the
hydrogen circle is drawn too small in comparison with
that of chlorine. If heat rays with all possible wave-

178 THE ATOM AND THE BOHR THEORY

lengths around 3-5 fb are sent through the hydrogen
chloride, that radiation energy will be absorbed which
can in part set the nuclei in oscillation and in part change
the state of rotation. Let us for a moment assume that
only the former change could happen. Then a ray with
wave-length 3*46 [L would be absorbed, this frequency
corresponding to the energy in the stationary state of
oscillation into which the molecule goes ; this frequency
is very nearly equal to the frequency with which the
nuclei vibrate relatively to each other. In reality, at
the same time that the nucleus is set in oscillation, there
will always be a change in the state of rotation — con¬
sisting either in an increase or in a decrease in the
velocity of rotation. The energy absorbed, and therefore
the frequency for the radiation absorbed, is thereby
changed a little, so that in the spectrum of the rays sent
through we do not obtain an absorption line correspond¬
ing to 3-46 [h, but a line somewhat removed from that.
Since there are, however, many stationary states of
rotation to start from, and since in some molecules there
is one transition, in others another, we get many absorp¬
tion lines on each side of 3-46

Even before Bohr propounded his theory, at a time
when the quantum theory did not yet have a clarified
form, the Danish chemist, Niels Bjerrum, had predicted
that the infra-red absorption lines ought to have such a
structure. This structure must be interpreted in the
above way which differs somewhat from Bjerrum's ideas,
but his prediction was essentially strengthened by
investigations, and it was one of the most significant
features in the development of the quantum theory
prior to 1913. The first to detect the structure of the
infra-red absorption bands was the Swedish physicist,

APPLICATIONS OF THE BOHR THEORY 179

Eva von Bahr. Her experiments were later extended in
a most significant way by the work of Imes and other
American investigators. They enable us to calculate
exactly the distance between the two nuclei in the
molecule.

It may be asked what becomes of the energy which
the hydrogen chloride molecule thus absorbs, and
whether it necessarily after a longer or shorter time
must be re-emitted as radiation. The latter is not the
case. In a collision between molecules or atoms, the
energy which one molecule (or atom) has absorbed by
radiation can undoubtedly be transferred to another
molecule, the velocity of which is thereby increased.
The theoretical necessity of the occurrence of such colli¬
sions was clearly shown for the first time in a very
significant investigation by two of Bohr’s students, Klein
and Rosseland. Without collisions of this nature the
radiation energy absorbed could never be transformed
into heat energy. Here we come to a very great and
important field, which has a very close connection with
the theory of the chemical processes and to a better ex¬
planation of which the more recent experiments of Franck
and his co-workers have made important contributions.

CHAPTER VII

THE STRUCTURE OF THE ATOM AND THE
CHEMICAL PROPERTIES OF THE ELEMENTS

Introduction.

We have hitherto restricted ourselves mainly to
those applications of the Bohr theory which have a
direct connection with the processes of radiation. We
have shown how fertile the theory has proved to be,
how many problems, previously inexplicable, have been
solved, and what exact agreement has been established
between experiment and theory in this comprehensive
field. We may now ask how the theory accounts for
the chemical behaviour of the different elements. As
early as 1913, Bohr, in connection with his researches
on spectral phenomena, had considered the chemical
properties of the elements and had pointed out interest¬
ing possibilities.

The Combination of Atoms into Molecules.

In his discussion of hydrogen, Bohr suggested a
model for the structure of its molecule, which we shall
give here, because, by a simple example, it illustrates
how two neutral atoms may form a molecule (cf. p. 13).
In Fig. 33, a, K1E1 and K2E2 are two neutral hydrogen
atoms which are approaching each other with the orbits

180

STRUCTURE OF THE ATOM 181

of the two electrons parallel. The nucleus K1 and the
electron E2 then attract each other as do the nucleus K2
and the electron Ev The two electrons repel each other
as do the two nuclei ; but when the electrons are in
opposite positions of their orbits, the forces of attraction
outweigh the effect of the forces of repulsion. Cal¬
culation shows that when the atoms are allowed to

Eig. 33. — Early representation of the formation of a hydrogen

molecule (Bohr, 1913).

approach each other the positions of the atoms will be
as is shown in Fig. 33, b, where the orbits are closer
to each other than the nuclei. Finally, for small
distances of the two nuclei, the two orbits will be merged
into one, as is shown in Fig. 33, c. This orbit will be
slightly larger than the original ones. The two hydrogen
atoms may, in this way, combine into one molecule.
In fact, an equilibrium position can be found for which

182 THE ATOM AND THE BOHR THEORY

the nuclei are held together, in spite of their mutual
repulsion, by the attractive forces existing between
them and the electrons which are moving in their
common orbit. It must, however, be assumed, for
reasons which cannot be given here, that the hydrogen
molecule is, in reality, constructed somewhat differently ;
probably the orbits of the electrons make an angle with
each other.

The formation of a hydrogen molecule may also be
supposed to occur when a positive hydrogen ion, i.e., a
hydrogen nucleus, and a negative hydrogen ion, i.e., a
hydrogen nucleus with two electrons, are drawn together
by their mutual attractions. The forces of attraction
would be much stronger than in the first example given,
and the formation of a neutral molecule would not take
place in the same way. More energy would also be
released, but the final result will be the same.

Just as in the case of the hydrogen molecule, other
molecules may be formed from atoms belonging to the
same or to different elements. The method of forma¬
tion of molecules varies according as it is a union of
neutral molecules in the normal state, or a union of
positive and negative ions. Conversely, by chemical
decomposition a molecule can be separated either into
neutral parts or into ions. If, for instance, common
salt (sodium chloride, NaCl) is dissolved in water, the
salt molecules are, under the influence of the water
molecules, decomposed in Na-ions with one positive
charge and Cl-ions with one negative charge, correspond¬
ing to the monovalent electropositive character of
sodium and the monovalent electronegative character of
chlorine.

The possibilities are, however, far from being ex-

STRUCTURE OF THE ATOM 183

hausted by these two methods of composition and de¬
composition. An atom may exist not only in the normal
state where it has its complete number of electrons
collected as tightly as possible about the nucleus, and
in the ionized state with one or more too many or too
few electrons ; but in a neutral atom one or more of the
outer electrons may be in a stationary orbit at a greater
distance from the nucleus than corresponds to the
normal stationary state. It is easy to understand that
an atom in such an “ excited ” or (as it is called in
chemistry) active state often finds it easier to act in
concert with other atoms than when it is in the normal
state ; in this latter state the atom is often more like a
little compact lump of neutral substance than in the
active state.

It will, in any case, be understood that the interplay
between the atoms, which reveals itself in the chemical
processes or reactions between different elements, offers
many opportunities for the Bohr theory to give in the
future a more detailed explanation than was possible to
the earlier theories of chemistry. We must also mention
the fact that it has become possible to elucidate in main
features the phenomena, hitherto unexplained, of the
chemical effects of light, as on a photographic plate
( photochemistry ) and of catalysis , which consists in
bringing about, or accelerating, the chemical inter¬
action between two substances by the presence of a
third substance which does not itself enter in the com¬
pound, and often needs only to be present in very small
quantities. It must, however, be emphasized that at
present we do not yet possess a detailed theory of
molecular constitution comparable with our knowledge
of the structure of the atoms,

184 THE ATOM AND THE BOHR THEORY

The Periodic System.

Instead of inquiring how the chemical processes
may take place, we shall now study the general correla¬
tion between the chemical properties and the atomic
numbers of the elements, a correlation which has found
its empirical expression in the natural or periodic system
of the elements (cf. p. 23). The explanation of the
puzzles of this system must be said to be one of the
finest results which Bohr has obtained, and it constitutes
a striking evidence in favour of the quantum theory of
atoms.

There is nothing new in the idea of connecting the
arrangement of the elements in the peroidic system with
an arrangement of particles in the atom in regular groups ,
the character of which varies, so to say, periodically
with increasing number of particles. In the atom model
of Lord Kelvin and j. j. Thomsom (cf. p. 86), with the
positive electricity distributed over the volume of the
whole atom, Thomson tried to explain certain leading
characteristics of the periodic system by imagining the
electrons as arranged in several circular rings about the
centre of the atom. He pointed out that the stability
of the electronic configurations of this type varied in a
remarkable periodic way with the number of electrons
in the atom. By considerations of this nature Thomson
was able to enunciate a series of analogies to the
behaviour of the elements in the periodic system as
regards the tendency of the neutral atoms to lose one or
more electrons (electropositive elements) or to take up
one or more electrons (electronegative elements). But,
setting aside possible objections to his considerations
and calculations, the connection with the system was

STRUCTURE OF THE ATOM

185

very loose and general, and his theory lost its funda¬
mental support when his atomic model had to give way
to Rutherford’s. With Bohr’s theory the demand for
a stable system of electrons was placed in an entirely
new light.

In his treatise of 1913, Bohr tried to give an explana¬
tion of the structure of the atom, by thinking of the
electrons as moving in a larger or smaller number of
circular rings about the nucleus. His theory did not
exclude the possibility of orbits of electrons having
different directions in space instead of lying in one plane
or being parallel. The tendency of the considerations
was to attain a definite, unique determination of the
structure of the atom, as is demanded by the pronounced
stability of the chemical and physical properties of the
elements. The results were, however, rather unsatis¬
factory, and it became more and more clear that the
bases of the quantum theory were not sufficiently
developed to lead in an unambiguous way to a definite
picture of the atom. Nowadays the simple conception
of the electrons moving in circular rings in the field of
the nucleus is definitely abandoned, and replaced by a
picture of atomic constitution of which we shall speak
presently.

In the following years the general conception of the
group distribution of the electrons in the atom formed the
basis of many theoretical investigations, which in various
respects have led to a closer understanding of chemical
and physical facts. The German physicist, Kossel,
showed that the characteristic X-ray spectra of the
elements, which are due to a process of reconstruction
of the atom subsequent to the removal of one or more of
the innermost electrons (cf. p. 161), give a most striking

./ '

186 THE ATOM AND THE BOHR THEORY .

jfm

support to the assumption that the electrons are dis¬
tributed in different groups in which they are bound with
different strength to the atom.

The connection between the electron groups and the
chemical valence properties of the atoms, to which
Thomson had first drawn attention and which al^o played
an important part in Bohr’s early considerations, was
further developed in a significant way by Kossel, as
well as by Lewis and by Langmuir in America. These
chemical theories had, however, little or no connection
with the quantum theory of atomic processes ; even the
special features of the Rutherford atom, which are of
essential importance in the theory of the hydrogen
spectrum and of other spectra, played only a subordinate

part.

In 1920 Bohr showed how, by the development of the
quantum theory which had taken place in the meantime,
and the main features of which consisted in the intro¬
duction of more than one quantum number for the
determination of the stationary states and in the estab¬
lishment of the correspondence principle, the problem
of the structure of the atom had appeared in a new light.
In fact, he outlined a general picture of atomic constitu¬
tion, based on the quantum theory, which in a remarkable
way accounted for the properties of the elements. In
order to decide doubtful questions, he has often had to
call to his aid the observed properties of elements, and it
must be readily admitted that the finishing touches of
the theory are still lacking. But from his general
starting-point he has been able to outline the architecture
of even the most complex atomic structures and to explain,
not only the known regularities, but also the apparent
irregularities of the periodic system of the elements,

STRUCTURE OF THE ATOM

187

The method Bohr used in his attempt to solve the prob¬
lem was to study how a neutral normal atom may gradually
be formed by the successive capturing and binding of the
individual electrons in the field of force about the nucleus of
the atom. He began by assuming that he had a solitary
nucleus with a positive charge of a given magnitude.
To this nucleus free electrons are now added, one after
the other, until the nucleus has taken on the number
sufficient to neutralize the nuclear charge. Each indi¬
vidual electron undergoes a “ binding ” process, i.e. it
can move in different possible stationary orbits about
the nucleus and the electrons already bound. With the
emission of radiation it can go from stationary states
with greater energy to others with less energy, ending
its journey by remaining in the orbit which corre¬
sponds to the least possible energy. We may designate
this state of least energy as the normal state of the
system, which, however, is only a positive atomic ion,
so long as all the electrons needed for neutralization are
not yet captured.

From the exposition in the preceding chapter it will
be seen that the ordinary series spectra (arc spectra) may
be considered as corresponding to the last stage in this
formation process, since the emission of each line in such
a spectrum is due to a transition between two stationary
states in each of which N-i electrons are bound in
their normal state, i.e. as tight as possible, by the nucleus,
while the Nth electron moves in an orbit mainly outside
the region of the other electrons. In the same way the
spark spectra give witness of the last stage but one of the
formation process of the atom, since here N-2 electrons
are bound in their normal state while an N-ith electron
moves in an orbit large compared with the dimensions of

188 THE ATOM AND THE BOHR THEORY

the orbits of the inner electrons. From these remarks
it will be clear that the study of the series spectra is of
great importance for the closer investigation of the
process of formation of the atom outlined above.
Furthermore, the general ideas of the correspondence
principle, which directly connects the possibility of
transition from one stationary state to another with the
motion of the electron, has been very useful in throwing
light on the individual capturing processes and on the
stability of the electronic configurations formed by
these. In what follows we cannot, however, reproduce
Bohr’s arguments at length ; we must satisfy ourselves
with some hints here and there, and for the rest
restrict ourselves to giving some of the principal
results.

Before going farther we shall recall what has pre¬
viously been said about the quantum numbers. In the
undisturbed hydrogen atom, the stationary orbits can
be numbered with the principal quantum numbers
i, 2, 3 ... n. But to each principal quantum number
there corresponds not one but several states, each with
its auxiliary quantum number i, 2, 3 . . . k, k at the most
being equal to the principal quantum number. In a
similar way, the stationary orbits of the electrons in an
atom containing several electrons can be indicated by
two quantum numbers, the 32 orbit, for instance, being
that with principal quantum number 3 and auxiliary
quantum number 2. But while in the hydrogen atom
the principal quantum number n, in the stationary
orbits which are slowly rotating ellipses, is very simply
connected with the length of the major axis of the
ellipse, and k : n is the ratio between the minor and
major axes, still in other atoms with complex systems

STRUCTURE OF THE ATOM 189

of electrons the significance of the principal quantum
number is not so simple and the orbit of an electron
consists of a sequence of loops of more complicated form
(cf. Fig. 29). We must satisfy ourselves with the
statement that a definition of their significance can be
given, but only by mathematical-physical considerations
which we cannot enter into here. It may, however, be
stated that, if we restrict ourselves to a definite atom,
the rule will hold that, among a series of orbits with
the same auxiliary quantum number but different princi¬
pal quantum numbers, that orbit in which the electron
attains a greater distance from the nucleus has the higher
number. Another rule which holds is, that an orbit with
a small auxiliary quantum number in comparison with
its principal quantum number (as 4X, for instance, cf.
Fig. 29), will consist of very oblong loops with a very
great difference between the greatest and least distances
of the electron from the nucleus, while the orbit will be a
circle when the two quantum numbers are the same as
for iv 22, 33. Although each orbit has two quantum
numbers, we often speak simply of the 1-, 2-, 3 - ... n-
quantum orbits, meaning here the orbits with the
principal quantum numbers 1, 2, 3 . . . n.

The one electron of hydrogen will, upon being
captured, first be at “ rest ” when it reaches the irpath,
and we might perhaps be led to expect that in the
atoms with greater nuclear charges the electrons in the
normal state also would be in the one quantum orbit iv
because to this corresponds the least energy in hydrogen.
This assumption formed actually the basis of Bohr’s
work of 1913 on the structure of the heavier atoms.
It cannot be maintained, however. Considerations of
theoretical and empirical nature lead to the assump-

190

THE ATOM AND THE BOHR THEORY

tion that the electrons which already are gathered
about the nucleus can make room only to a certain extent
for new ones, moving in orbits of the same principal
quantum number. Those electrons which are captured
later are kept at an appropriate distance ; they are, for
instance, prevented from passing from a 3-quantum
orbit to a 2-quantum one, if the number of electrons
moving in 2-quantum orbits has reached a certain
maximum value. When it is said that the captured
electrons end in the stationary state which corresponds
to the least energy, it must, therefore, mean, not the
i-quantum orbit, but the innermost possible under the
existing circumstances. The final result will be that the
electrons are distributed in groups , which are characterized
each by their quantum numbers in such a way that passing
from the nucleus to the surface of the atom , the successive
groups correspond to successive integer values of the
quantum number , the innermost group being characterized
by the quantum number one. Moreover , each group is
subdivided into sub-groups corresponding to the different
values which the auxiliary quantum number may
take.

That the electrons first collected keep the late¬
comers at an appropriate distance must be understood
with reservations ; a new electron moving in an elon¬
gated orbit can very well come into the territory already
occupied ; in fact, it may come closer to the nucleus
than some of the innermost groups of electrons. In case
an outer electron thus dives into the inner groups, it
makes a very short visit, travelling about the nucleus
like a comet which at one time on its elongated orbit
comes in among the planets and perhaps draws closer
to the sun than the innermost planet, but during the

STRUCTURE OF THE ATOM

191

greater part of its travelling time moves in distant
regions beyond the boundaries of the planetary system.
It is a very important characteristic of the Bohr theory
of atomic architecture that the outer electrons thus
penetrate far into the interior of the atom and thus
chain the whole system together.

Such a “ comet electron ” has, however, a motion of
a very different nature from that of a comet in the solar
system. Let us suppose that the nuclear charge is 55
(Caesium), that there already are fifty-four electrons
gathered tightly about the nucleus, and that No. 55 in
an orbit consisting of oblong loops moves far away from
the nucleus, but at certain times comes in close to it.
Then, for the greater part of its orbit, this electron will
be subject to approximately the same attraction as the
attraction towards one single charge, as a hydrogen
nucleus ; but when No. 55 comes within the fifty-four
electrons it will for a very short time be influenced by
the entire nuclear charge 55. Together with the near¬
ness of the nucleus, this will cause No. 55 to acquire a
remarkably high velocity and to move in an orbit quite
different from the elliptical one it followed outside.
Moreover, the great velocity of the electron during its
short visit to the nucleus is in a considerable degree
determinative of its principal quantum number ; this
will be higher than would be expected from the
dimensions of the outer part of the orbit if we supposed
the motion to take place about a hydrogen nucleus
(cf. Figs. 27 and 29).

After these general remarks we shall try in a few
lines to sketch the Bohr theory of the structure of the
atomic systems from the simplest to the most compli¬
cated. We shall not examine the entire periodic system

192 THE ATOM AND THE BOHR THEORY

with its ninety-two elements, but here and there we shall
bring to light a trait which will illustrate the problem —
partly in connection with the schematic representations
in the atomic diagrams at the end of the book.

Description of the Atomic Diagrams.

The curves drawn represent parts of the orbital
loops of the electrons in the neutral atoms of different
elements. Although the attempt has been made to give
a true picture of these orbits as regards their dimensions,
the drawings must still be considered as largely sym¬
bolic. Thus in reality the orbits do not lie in the same
plane, but are oriented in different ways in space. It
would have been impracticable to show the different
planes of the orbits in the figure. Moreover, there is
still a good deal of uncertainty as to the relative positions
of these planes. On this account the orbits belonging
to the same sub-group, i.e., designated by the same
quantum numbers, are placed in a symmetric scheme in
the sketch. For groups of circular orbits the rule has
been followed to draw only one of them as a circle,
while the others in the simpler atoms are drawn in pro¬
jection as ellipses within the circle, and in the more
complicated atoms are omitted entirely. The two
circular orbits of the helium atom are both drawn in
projection as ellipses. Further, for the sake of clear¬
ness, no attempt has been made to draw the inner loops
of the non-circular orbits of electrons which dive into
the interior of the atom. In lithium only, the inner
loop of the orbit of the 2X electron has been shown by
dotted lines.

In order to distinguish the groups of orbits with
different principal quantum numbers two colours have

STRUCTURE OF THE ATOM 193

been used, red and black, the red indicating the
orbits with uneven quantum numbers, as i, 3, 5, the
black those with even quantum numbers, as 2, 4, 6.

. Wherever possible the nucleus is indicated by a black
dot ; but in the sketches of atoms with higher atomic
numbers the i-quantum orbits are merged into one
little cross and the nucleus has been omitted. It should
be noticed that the radium atom is drawn on a scale
twice as great as that for the other atoms

We shall begin with the capture of the first electron.
If the nucleus is a hydrogen nucleus the hydrogen atom
is completed when the electron has come into the i±
orbit, a circle with diameter of about io~8 cm. (cf. the
diagram) . If the nucleus had had a greater nuclear charge
the No. 1 electron would have behaved in the same way,
but the radius of its orbit would have been less in the
same ratio as the nuclear charge was greater. For a
lead nucleus, with charge (atomic number) 82, the radius
of the ix orbit is 1/82 that of the hydrogen nq orbit.
Since atoms with high atomic numbers thus collect the
electrons more tightly about them it is understandable
that, in spite of their greater number of electrons, they can
be of the same order of magnitude as the simpler atoms.

Let us now examine the helium atom. The first
electron, which its nucleus (charge 2) catches, moves as
shown in a circle ilf but with a smaller radius than in
the case of the hydrogen atom. Electron No. 2 can be
caught in different ways, and the closer study of the
conditions prevailing here, which are still comparatively
simple since there are only two electrons, has been of
greatest importance in the further development of the
whole theory. We cannot go into it here, but must
r3

194

THE ATOM AND THE BOHR THEORY

content ourselves with saying that the stable final
result of the binding of the second electron consists m
the two electrons moving in circular i-quantum orbits of
the same size with their planes making an angle with
each other (cf. the diagram). This state has a very stable
character, and the helium atom is therefore very dis¬
inclined to interplay with other atoms, with other
helium atoms as well as with those of other elements.
Helium is therefore monatomic and a chemically inactive

gas.

In all atomic nuclei with higher charges than the
helium nucleus the first two orbits are also bound into
two i-quantum circular orbits at an angle with each
other; this group cannot take up any new electron
having the same principal quantum number. It takes
on an independent existence and forms the innermost
electron group in all atoms of atomic number higher

than 2. . ,

Electron No. 3 will accordingly not be bound m the

same group with 1 and 2. It must be satisfied with a
2-quantum orbit, 2V which consists of oblong loops, and,
when nearest the nucleus, comes into the territory of
the i-quantum orbits. It is but loosely bound com¬
pared to the first two electrons, and the lithium atom,
which has only three electrons, can therefore easily let
No. 3 loose so that the atom becomes a positive ion.
Lithium is therefore a strongly electropositive monovalent
metal. The element beryllium (No. 4) will probably
have two electrons in the orbits 2t ; it will therefore be
divalent. But during the short visit of these electrons
to the nucleus they are subject to a greater nuclear
charge than in lithium. The 2X electrons are therefore,
in beryllium, more firmly bound than in lithium, and the

STRUCTURE OF THE ATOM 195

electropositive character of beryllium is therefore less
marked.

\ .

We have something essentially new in the boron
atom (atomic No. 5) where the two electrons No. 3 and
No. 4 are taken into 2X orbits, but where No. 5 will very
probably be bound in a circular 22 orbit. How the
conditions will be in the normal state of the following
atoms preceding neon is not known with certainty. We
only know that the electrons coming after the first two
will be captured in 2-quantum orbits, the dimensions of
which get smaller, according as the atomic number
increases.

The neon atom (compare the diagram) has a parti¬
cularly stable structure, both " closed ” and symmetric,
which besides two 34-orbits contains four electrons in
2X orbits and four electrons in 22 orbits. As regards the
four electrons in 21 orbits, they do not have symmetri¬
cal positions at every moment or move simultaneously
either towards or away from the nucleus ; on the contrary,
it must be assumed that the electrons come closest to the
nucleus at different moments at equal intervals of time.

The name of inert or inactive gases is given to the
entire series of helium (2), neon (10), argon (18), krypton
(36), xenon (54) and niton (86), the O-column in the
periodic system given in the table on p. 23. All these
elements are monatomic and quite unwilling to enter into
chemical compounds with other elements (although there
is about 1 per cent, of argon in the air about us this
element has, on this account, escaped the observation
of chemists until about 1895, when it was discovered by
the English chemist, Ramsay). This complete chemical
inactivity is explained by the fact that the atoms of all
these elements have a nicely finished “ architecture ”

196 THE ATOM AND THE BOHR THEORY

Table showing the Distribution of the Electrons
OF DIFFERENT ORBITAL TYPES IN THE NEUTRAL

Atoms of the Inactive Gases.

Atomic

Number.

- - - -

Quantum Numbers.

1 1

3s 4i

5’

6«

7,1

—

Helium . .

Neon . . .

Argon . . .

Krypton . .

Xenon . . .

Niton . . .

118

—

with all the electrons firmly bound in symmetrical con¬
figurations. They may be said to form the mile posts of
the periodic system, and to be the ideals towards which
the other atoms aspire. The table shows how the
electrons in the atoms of these gases are divided among
the types of orbits corresponding to the different quantum

numbers.

The elements fluorine, oxygen and nitrogen can attain
the ideal neon-architecture by binding respectively one,
two and three additional electrons. Naturally they do
not become neon atoms, but merely negative atomic
ions with single, double or triple charge; and their
tendency in this direction appears in their character of
monovalent, divalent and trivalent electronegative
elements respectively. If we return to carbon it can
probably not become a tetravalent negative ion by
binding four free electrons ; but in the typical carbon
compound, methane (CH4), the neon ideal is realized in
another manner. In fact, it is reasonable to assume that
the four electrons of the hydrogen atoms together with
the six of the carbon atom give approximately a neon

STRUCTURE OF THE ATOM 197

architecture. The four hydrogen nuclei naturally cannot
be combined with the carbon nucleus ; the mutual re¬
pulsions keep them at a distance. They will probably
assume very symmetrical positions within the electron
system which holds them together. The nitrogen atom
may in a similar way find completion in a neutral mole¬
cule with neon-architecture, if it unites with three
hydrogen atoms to form ammonia NH3 ; but the three
hydrogen nuclei, although having symmetrical positions,
will not lie in the same plane as the nitrogen nucleus.
The electric centre of gravity for the positive nuclei will
therefore not coincide with the centre of gravity for the
negative electron system. The molecule obtains thus
what might be called a positive and a negative pole,
and this dipolar character appears in the electrical
action of ammonia (its dielectric constant). Something
similar holds true for the water molecule, where, in a
neon-architecture of electrons, in addition to the oxygen
nucleus in the centre there are two hydrogen nuclei
which are not co-linear with the oxygen nucleus.

If we go on from neon in the periodic system we come
to sodium (n). When the sodium nucleus captures
electron No. n, this cannot find room in the neon-
architecture formed by the first ten electrons. Since
the eleventh electron thus cannot find a place in either a
2X or a 22 orbit, it is bound in a 3X orbit (cf. Fig. 29 and
diagram at the end) . The atom then has a character like
that of the lithium atom, and we can therefore under¬
stand the chemical relationship between the two elements,
which are both monovalent electropositive metals.

We shall not dwell longer upon the individual elements
of the atomic series. If we pass from neon through
sodium (11), magnesium (12), aluminium (13), etc., to

198 THE ATOM AND THE BOHR THEORY

argon (18), we get what is essentially a repetition of the
situation in the series from lithium to neon. We first
get two orbits of the type in magnesium, a 32 orbit is
for the first time added in aluminium, and for the atomic
number 18, eight 3-quantum orbits, together with the
eight orbits of the inner 2-quantum group and the two of
the innermost i-quantum group, give the symmetric
architecture of argon (cf. table on p. 196, and diagram at
the end) .

The architecture of the argon atom is in a certain
sense less complete than that of the neon atom. In
argon there are indeed four orbits of the 3X type and four
of the 32 type, but the third kind of 3-quantum orbit, the
circular 33 one, is missing. Nor does it appear in the
next element, potassium (19). The electron No. 19
prefers, instead of the 3S orbit, the 4X orbit, which consists
of oblong loops and which gives a firmer binding because
it dives in among the electrons bound earlier, while the
circle 33 would lie outside them all. We thus obtain an
atom of type similar to the lithium and sodium atoms.
But the slighted 33 path lies, so to speak, on the watch
to steal a place for itself in the neutral atom, and this
has grave results for the subsequent development.
Even in calcium (20), after the first eighteen electrons are
bound in the argon architecture, both the nineteenth and
the twentieth go into a 41 orbit, and the behaviour of
calcium is like that of magnesium. But since the in¬
creasing nuclear charge means for the electron No. 19
a decrease in the dimensions and an increase in the bind¬
ing of the orbits corresponding to the quantum number
33, a point will finally be attained where the 33 orbit of
the nineteenth electron lies within the boundaries of what
may be called the argon system, i.e., the architecture

STRUCTURE OF THE ATOM 199

corresponding to the first eighteen electrons, and corre¬
sponds to a stronger binding than a 4i orbit would do.
In scandium (21) the 33~tyPe orbit occurs for the first
time in the neutral atom and will not only come into
competition with the 4i type, but will also cause a disturb¬
ance in the 3-quantum groups, which in the following
elements must undergo reconstruction. As long as this
lasts the situation is very complicated and uncertain.
When the reorganization is almost completed, we come
to the blotting out of chemical differences, particularly
known from the triad, iron, cobalt and nickel. More¬
over, there comes a fluctuation in the valency of
the elements. Iron can, as has been said, be divalent,
tri valent or hexa valent. This oscillation in valency
begins in titanium.

We should perhaps expect that the reconstruction
would be completed long before nickel (28) is reached,
because even with twenty-two electrons we could get
four orbits of each of the 3-quantum types (3X, 32 and
33) ; but from the chemical facts we are led to assume
that in a completed group of 3-quantum orbits there
can be room for six electrons in each sub-group. At
first sight we should, then, expect the end of the re¬
construction with nickel, which has indeed eighteen
electrons more than neon where the group of 2-quantum
orbits was completed. We might expect that nickel
would be an inert element in the series with helium, neon,
and argon. On the contrary, nickel merely imitates
cobalt. This is explained by the fact that the group of
eighteen 3-quantum orbits, although it has a symmetric
architecture, is weakly constructed if the nuclear charge
is not sufficiently large. The binding of this group is
too weak for it to exist as the outer group in a neutral

200 THE ATOM AND THE BOHR THEORY

atom. In nickel the electrons, in a less symmetical
manner, will probably arrange themselves with seventeen
3-quantum orbits and one 4-quantum orbit.

The group of eighteen 3-quantum orbits becomes
stable, however, when the nuclear charge is equal to or
larger than 29, in which case it can become the outer
group in a positive ion. In this we find the explanation
of the properties of the atom of copper. The neutral
copper atom has its twenty-ninth electron bound in a
4X orbit consisting of oblong loops (cf . diagram at the end) ;
this electron can easily be freed and leaves a positive
monovalent copper ion with a symmetrical architecture.
Even under these circumstances, although possessing a
certain stability, the ion is not very firmly constructed.
Thus the fact that copper can be both monovalent and
divalent, must be explained by the assumption that for
a nuclear charge 29, the 3-quantum group still, easily
loses an electron.

When we come to zinc (30) the group of eighteen is
more firmly bound ; zinc is a pronounced divalent metal
which in its properties reminds us of calcium and mag¬
nesium. From zinc (30) to krypton (36) we have a
series of elements which in a certain way repeat the series
from magnesium (12) to argon (18).

In Fig. 34 is shown Bohr’s arrangement of the
periodic system in which the systematic correlation of
the properties of the element appears somewhat clearer
than in the usual plan (cf. p. 23). It shows great
similarity with an arrangement proposed nearly thirty
years ago by the Danish chemist, Julius Thomsen. The
elements from scandium to nickel, where, in the neutral
atom, the electron group of 3-quantum orbits is in a
state of reconstruction, are placed in a frame ; the

STRUCTURE OF THE ATOM

201

Fig. 34. — The periodic system of the elements. The elements where
an inner group of orbits is in a stage of reconstruction are
framed. The oblique lines connect elements which in physical
and chemical respects have similar properties.

202 THE ATOM AND THE BOHR THEORY

oblique lines connect elements which are “homologous,”
i.e., similar in chemical and physical (spectral) respect.

In krypton (36) we again have a stable architecture
with an outer group of eight electrons, four in 4 orbits and
four in 4 orbits. Owing to the appearance of 43 orbits
in the normal state of the atoms of elements with atomic
number higher than 38, there follows in the fifth period
of the natural system a reconstruction and provisional
completion of the 4-quantum orbits to a group of eighteen
electrons, which shows a great simplicity with the comple¬
tion of the 3-quantum group in the fourth period. In Fig.
34 the elements where the 4-quantum group is in a state of
reconstruction are framed. The 4-quantum group with
eighteen electrons is of more stable construction than the
group of eighteen 3-quantum orbits in the elements with
an atomic number lower by eighteen. This is due to the
fact that all the orbits in the first-mentioned group are
oblong and therefore moored, so to say, in the inner
groups, while in the complete group of 3-quantum
orbits there are six circular orbits. This is the reason
why silver, in contrast to copper, is monovalent.

The next inactive gas is xenon (54), which outside of
the 4-quantum group has a group of eight electrons in
5-quantum orbits, four in 5X orbits and four in 52 orbits.
We notice that in xenon the group of 4-quantum orbits
still lacks the 44 orbits. On the theory we must, there¬
fore, expect to meet a new process of completion and
reconstruction when proceeding in the system of the
elements. The theoretical argument is similar to that
which applies in the case of the completion of the 3-
quantum group which takes place in the fourth period
of the natural system. In fact, in the formation of
the normal atoms of the elements next after xenon,

STRUCTURE OF THE ATOM 203

caesium, 55, and barium, 56, the fifty-four electrons first
captured will form a xenon configuration, while the
fifty- fifth electron will be bound in a 6X orbit, consisting
of very oblong loops, which represents a much stronger
binding than a circular 44 orbit. Calculation shows,
however, that with increasing nuclear charge there must
soon appear an element for which a 44 orbit will re¬
present a stronger binding than any other orbit. This
is actually the case in cerium (58), and starting from this
element we meet a series of elements where, in the
normal neutral atom, the 4-quantum group is in a state
of reconstruction. This reconstruction must occur far
within the atom, since the group of eighteen 4-quantum
orbits in xenon is already covered by an outer group
of eight 5-quantum orbits. The result is a whole series
of elements with very slight outward differences between
their neutral atoms, and therefore with very similar
properties. This is the rare earths group, which in such
a strange way seemed to break down the order of the
natural system (cf. p. 21), but which thus finds its natural
explanation in the quantum theory of the structure of
the atom.

The elements in which the 4-quantum group is in a
state of reconstruction are, in Fig. 34, enclosed in the
inner frame in the sixth period. Moreover, in the outer
frame all elements are enclosed where the group of
5-quantum orbits is in a state of reconstruction, which
started, even before cerium in lanthanum (57), where
the fifty-fifth electron in the normal state is bound in a
53 orbit. The element cassiopeium, with atomic number
71, which is the last of the rare earths, stands just outside
the inner frame, because in the normal neutral atom of
this element the 4-quantum group is just completed ;

204 THE ATOM AND THE BOHR THEORY

this group, instead of eighteen electrons with six electrons
in each sub-group, consists now of thirty-two electrons
with eight electrons in each sub-group. The theory
was able to predict that the element with atomic number
72, which until a short time ago had never been found,
and the properties of which had been the subject of
some discussion, must in its chemical properties differ
considerably from the trivalent rare earths and show a
resemblance to the tetra valent elements zirconium (40),
and thorium (90). This expectation has recently been
confirmed by the work of Hevesy and Coster in Copen¬
hagen, who have observed, by means of X-ray investiga¬
tions, that most zircon minerals contain considerable
quantities (1 to 10 per cent.) of an element of atomic
number 72, which has chemical properties resembling
very much those of zirconium, and which on this account
had hitherto not been detected by chemical investigation.
A preliminary investigation of the atomic weight of this
new element, for which its discoverers have proposed the
name hafnium (Hafnia= Copenhagen), gave values lying
between 178-180, in accordance with what might be
expected from the atomic weight of the elements (71)
and (73). (Cf. p. 23.)

The further completion of the groups of 5- and 6-
quantum orbits, which in the rare earths had tempor¬
arily come to a standstill, is resumed in hafnium and
goes on in a way very similar to that in which the 4- and
5-quantum groups in the fifth, and the 3- and 4-quantum
groups in the fourth period underwent completion.
Thus the reconstruction of the 5-quantum group which
began in lanthanum, and which receives a characteristic
expression in the triad of the platinum metals, has come
to a provisional conclusion in gold (79), gold being the

205

STRUCTURE OF THE ATOM

first element outside the two frames which, in Fig. 34>
appear in the sixth period. The neutral gold atom
possesses, in its normal state — besides two i-quantum
orbits, eight 2-quantum orbits, eighteen 3-quantum
orbits, thirty-two 4-quantum orbits and eighteen
5-quantum orbits — one loosely bound electron in a 6^

orbit.

In niton (86), finally, we meet again an inactive gas,
the structure of the atom of which is indicated in the

table on p. 196.

In this element the difference between nuclear and
electron properties appears very conspicuously, since
the structure of the electron system is particularly
stable, while that of the nucleus is unstable. Niton, in
fact, is a radioactive element which is known in three
isotopic forms ; one of these is the disintegration product
of radium, the so-called radium emanation ; it then has
a very brief life. In the course of four days over half of
the nuclei in a given quantity of radium emanation will

explode.

In the diagram at the end of the book, as an example
of an atom with very complicated structure, there is given
a schematic representation of the atom of the famous
element radium , on a scale twice as large as the one used
in the other atoms. It follows clearly enough, from what
has been said in Chap. IV., that the structure of the elec¬
tron system has nothing to do with the radioactivity.
All the remarkable radiation activities are due to the
nucleus itself. There has not even been room in the
figure to draw the nucleus ; the i-quantum orbits
consist only of a small cross, and in the other groups
we have contented ourselves with summary indications.
The electron system, with its eighty-eight electrons, is,

206 THE ATOM AND THE BOHR THEORY

however, in itself very interesting, with its symmetry in
the number of electrons in the different groups. In
the different quantum groups from i to 7 there are found
respectively two, eight, eighteen, thirty-two, eighteen,
eight and two electrons. The last group is naturally
of a very different nature from the first ; they are
‘‘valence electrons,” which easily get loose and leave
behind a positive radium ion with stable niton
architecture. Radium then belongs to the family of
the divalent metals, magnesium, calcium, strontium
and barium.

Four places from radium is uranium (92) and the end
of the journey, if we restrict ourselves to the elements
which are known to exist. One could very well continue
the building-up process still further and discuss what
structure would have to be assumed for the atoms of the
elements with higher atomic numbers. That they can¬
not exist is not the fault of the electron system but of
the nuclei, which would become too complicated and too
large to be stable. In the table on p. 196 there is shown
the probable structure of the atom of the inert gas
following niton ; it must be assumed to have one hundred
and eighteen electrons distributed in groups of two,
eight, eighteen, thirty-two, thirty-two, eighteen and
eight among the quantum groups from 1 to 7.

As has been said, in all this symmetrical structure of
the atoms of the elements, Bohr has in many cases had
to rely upon general considerations of the information
that observation gives about the properties of the
individual elements. It must, however, not be forgotten
that the backbone of the theory is and remains the
general laws of the quantum theory, applied to the
nucleus atom in the same way as they originally were

STRUCTURE OF THE ATOM 207

applied to the hydrogen atom, leading thereby to the
interpretation of the hydrogen spectrum.

We have, further, a most striking evidence as to the
correctness of Bohr’s ideas in the fact that not only do
the pictures of the atoms which he has drawn agree
with the known chemical facts about the elements, but
they are also able to explain in the most satisfactory
manner possible the most essential features of the
characteristic X.-ray spectra of the different elements, a
field we shall not enter upon here.

In all that has been said above we have been con¬
sidering the Bohr theory simply as a means of gaining
a deeper understanding of the laws which determine
activities in the atomic world. Perhaps we shall now
be asked if we can “ utilize ” the theory, or, in other

words, if it can be put to practical use.

To this natural and not unwarranted question we
may first give the very general answer, that progress in
our knowledge of the laws of nature always contributes
sooner or later, directly or indirectly, to increase our
mastery over nature. But the connection between
science and practical application may be more or less
conspicuous, the path from science to practical applica¬
tion more or less smooth. It must be admitted that
the Bohr theory, in its present state of development,
hardly leads to results of direct practical application.
But since it shows the way to a more thorough under¬
standing of the details in a great number of physical and
chemical processes, where the peculiar properties of the
different elements play parts of decided importance,
then in reality it offers a wealth of possibilities for making

208 THE ATOM AND THE BOHR THEORY

predictions about the course of the processes — predic¬
tions which undoubtedly in the course of time will be of
practical use in many ways. In this connection the dis¬
covery of the element hafnium, discussed on p. 204, may
be mentioned. It must be for the future to show what
the Bohr theory can do for technical practice.

Below is given an explanation of the different symbols

which occur at various places in the book ; also the values

of important physical constants.

i m. = i metre ; i cm. = i centimetre =0-394 inches.

1 fj. = i mikron =1/1000 of a millimetre =0-0001 cm. =10 4 cm.

1 up =1/1,000,000 of a millimetre = 1 o-7 cm.

1 cm.3 = 1 cubic centimetre.

1 g. = 1 gram ; 1 kg. = 1 kilogram = 2-2 pounds.

1 kgm. = 1 kilogrammetre (the work or the energy required
to lift 1 kg. 1 m.).

1 erg = 1-02 x io"8 kgm. =7-48 x io 8 foot-pounds.

X represents wave-length.

v represents frequency (number of oscillations in 1 second).

<0 represents frequency of rotation (number of rotations in
1 second).

n represents an integer (particularly the Bohr quantum
numbers) .

The velocity of light is c =3 X io10 cm. per second =9-9 x io8
feet per second.

The wave-length of yellow sodium light is 0-589 /x =589 hlx —
2-32 x io'5 inches.

The frequency of yellow sodium light is 526 x io12 vibrations
per second.

The number of molecules per cm.3 at o° C. and atmospheric
pressure is about 27 X io18.

The number of hydrogen atoms in 1 g. is about 6-io23.

The mass of a hydrogen atom is 1-65 X io-24 g.

The elementary quantum of electricity is 4-77 x io10 “ electro¬
static units.”

210 THE ATOM AND THE BOHR THEORY

The negative electric charge of an electron is i elementary
quantum (i negative charge).

The positive electric charge of a hydrogen nucleus is i ele¬
mentary quantum (i positive charge).

The mass of an electron is 1/1835 that of the hydrogen atom.

The diameter of an electron is estimated to be about 3 x
10 13 cm.

The diameter of the atomic nucleus is of the order of magni¬
tude 10 13 to io~12 cm.

The diameter of a hydrogen atom in the normal state Rhe
diameter of the first stationary orbit in Bohr’s model) is
1-056 x io-8 cm.

The Balmer constant K =3-29 x io15.

The Planck constant h= 6-54 x io'27.

An energy quantum is E = hv.

The Balmer-Ritz formula for the frequences of the lines in
the hydrogen spectrum is

Thallium

Sodium

Barium

STRUCTURE OF THE RADIUM ATOM

1;; r '

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RraaierS^n.A. & Holst, H.

541.2

K89

Th* atom & the Bohr theory of

its structure.

1923.

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